On the Geometry and Analysis of Graphs...On the Geometry and Analysis of Graphs Habilitationsschrift vorgelegt am 19. 08. 2014 der Fakult at fur Mathematik und Informatik der Friedrich-Schiller-Universit

On the Geometry andAnalysis of Graphs

Habilitationsschrift

vorgelegt am 19. 08. 2014

der Fakultat fur Mathematik und Informatik

der Friedrich-Schiller-Universitat Jena

von Dr. Matthias Keller

geboren am 31. 12. 1980 in Karl-Marx-Stadt, jetztChemnitz

Gutachter:Prof. Dr. Alexander Grigor’yan, Universitat BielefeldProf. Dr. Jurgen Jost, Max-Planck-Institut LeipzigProf. Dr. Daniel Lenz, Friedrich-Schiller-Universitat JenaProf. Dr. Karl-Theodor Sturm, Universitat Bonn

Vorgelegt am:

Erteilung der Lehrbefahigung am:

Acknowledgements

The submission of a habilitation thesis poses a landmark in aca-demic life which puts me in the privileged position to look back, sumup and, first and foremost, to express my gratitude to the various peo-ple that shared an active part during this time.

The first acknowledgement is due to Daniel Lenz who supportedme as an advisor, teacher, coauthor and friend during these years.His continuous support, inspiration, guidance and encouragement area corner stone of the present work. Furthermore, this work wouldnot have been possible without the expertise of my further coauthorsFrank Bauer, Michel Bonnefont, Sylvain Golenia, Sebastian Haeseler,Bobo Hua, Xueping Huang, Jun Masamune, Norbert Peyerimhoff andRadoslaw Wojciechowski.

This research was pursued at the Friedrich Schiller University Jenaand the Hebrew University Jerusalem. In Jena I am indebted tothe very productive, friendly and joyful atmosphere in the analysisgroup which would not be the same without my dear colleagues andfriends Siegfried Beckus, Felix Pogorzelski and last but not least MarcelSchmidt. In Jerusalem, I was blessed with the welcoming, open andvibrant environment of the Hebrew University and, in particular, thegenerous hospitality of my hosts Jonathan Breuer, who I gladly countamong my coauthors, and Dan Mangoubi.

During the past years I had the luxury to enjoy the hospitalityof various institutions and colleagues. In particular, I am glad tothank Alexander Grigor’yan (Bielefeld University), Jurgen Jost (MPILeipzig), Wolfgang Woess (Graz University of Technology), Peter Stoll-mann and Ivan Veselic (Chemnitz University of Technology), JozefDodziuk (Graduate Center CUNY, NYC), Alexander Teplayev andMichael Hinz (University of Connecticut), Shing-Tung Yau and GaborLippner (Harvard), Balint Virag (University of Toronto) and NabilaTorki-Hamza (Universite de Carthage) for the invitations and the prof-itable and pleasant time.

I was supported in this research by various grants and institutions. Igratefully acknowledge the support of the German Science foundationfor the project “Geometry of discrete spaces and spectral theory ofnon-local operators,” the Golda Meir Fellowship, the Israel ScienceFoundation (grant No. 1105/10 and No. 225/10) and BSF grant No.2010214. I am further thankful that Birkhauser, Springer, the London

1

2 ACKNOWLEDGEMENTS

Mathematical Society, Elsevier and Mathematical modeling of naturalphenomena granted their permission to reproduce the original articleswithin the thesis.

My parents, sisters and friends were always a source of strong sup-port during these years and I am thankful that I always found an openear. My life would not be half as exciting without my wonderful chil-dren Elliott, Lumen and Ferris who I cherish for being cheerful at daytime and sleeping peacefully at night time. Finally, the love and sup-port of my wife Yvonne can not be measured or expressed in wordsand I am endlessly grateful for the life we are sharing.

Contents

Acknowledgements 1

Structure of the work 7

Part 1. Introduction 9

Synopsis 11

Notation 15

Chapter 1. Dirichlet forms on discrete spaces 171.1. Graphs and Laplacians 171.2. The heat equation 231.3. Uniformly positive measure 261.4. Weakly spherically symmetric graphs 281.5. Sparseness 31

Chapter 2. Intrinsic metrics 352.1. Definition and basic facts 362.2. Liouville theorems 402.3. Domain of the generators and essential selfadjointness 432.4. Isoperimetric constants and lower spectral bounds 452.5. Volume growth and upper spectral bounds 482.6. Volume growth and `p-independence of the spectrum 51

Chapter 3. Curvature on planar tessellations 533.1. Set up and definitions 543.2. Curvature and the bottom of the spectrum 573.3. Decreasing curvature and discrete spectrum 613.4. The `p spectrum 633.5. Curvature on planar graphs 63

Bibliography 67

Part 2. Original Manuscripts 73

Chapter 4. M. Keller, D. Lenz, Dirichlet forms and stochasticcompleteness of graphs and subgraphs, Journal furdie reine und angewandte Mathematik 2012 (2012),189–223. 75

3

4 CONTENTS

Chapter 5. M. Keller, D. Lenz, Unbounded Laplacianson Graphs: Basic Spectral Properties and theHeat Equation, Mathematical modeling of naturalphenomena: Spectral Problems 5 (2010), 198–224. 111

Chapter 6. M. Keller, D. Lenz, R. Wojciechowski, Volumegrowth, spectrum and stochastic completeness ofinfinite graphs, Mathematische Zeitschrift 274(2013), 905–932. 139

Chapter 7. M. Bonnefont, S. Golenia, M. Keller, Eigenvalueasymptotics for Schrodinger operators on sparsegraphs, arXiv:1311.7221. 169

Chapter 8. B. Hua, M. Keller, Harmonic functions of generalgraph Laplacians, Calculus of Variations and PartialDifferential Equations 51 (2014), 343–362. 193

Chapter 9. X. Huang, M. Keller, J. Masamune, R. Wojciechowski,A note on self-adjoint extensions of the Laplacianon weighted graphs, Journal of Functional Analysis265 (2013), 1556–1578. 215

Chapter 10. F. Bauer, M. Keller, R. Wojciechowski, Cheegerinequalities for unbounded graph Laplacians, toappear in Journal of the European MathematicalSociety. 239

Chapter 11. S. Haeseler, M. Keller, R. Wojciechowski, Volumegrowth and bounds for the essential spectrum forDirichlet forms, Journal of the London MathematicalSociety 88 (2013), 883–898. 255

Chapter 12. F. Bauer, B. Hua, M. Keller, On the lp spectrumof Laplacians on graphs, Advances in Mathematics248 (2013), 717-735. 273

Chapter 13. M. Keller, The essential spectrum of the Laplacianon rapidly branching tessellations, MathematischeAnnalen 346 (2010), 51–66. 293

Chapter 14. M. Keller, N. Peyerimhoff, Cheeger constants,growth and spectrum of locally tessellating planargraphs, Mathematische Zeitschrift 268 (2011),871–886. 311

Chapter 15. M. Keller, Curvature, geometry and spectralproperties of planar graphs, Discrete &Computational Geometry 46 (2011), 500–525. 329

CONTENTS 5

Zusammenfassung in deutscher Sprache 357

Lebenslauf 361

Schriftenverzeichnis 371

Ehrenwortliche Erklarung 375

Structure of the work

The present work is divided into two parts. In the introductorypart, the main results of this work are presented; the second part con-sists of original manuscripts. The overall theme is the connection of ge-ometry, analysis and probability on discrete spaces modeled by Dirich-let forms. The geometric concepts under investigation are distance andcurvature and the derived consequences range from Liouville theorems,essential selfadjointness to spectral theory of selfadjoint operators.

The first part is divided into three chapters:

1. Dirichlet forms on discrete spaces [KL12, KL10, KLW13,BGK13].

2. Intrinsic metrics [HK14, HKMW13, BKW14, HKW13,BHK13].

3. Curvature on planar tessellations [Kel10, KP11, Kel11] (andalso [BGK13, BHK13]).

The exposition of the first and the second chapter will be partiallypublished in a slightly modified form in the survey article [Kel14b] andthe last chapter in the survey article [Kel14a]. The references listedafter the topics of the chapters above refer to the original manuscriptsthat form the second part of this thesis. Theses references are listedbelow:

[KL12] M. Keller, D. Lenz, Dirichlet forms and stochastic complete-ness of graphs and subgraphs, Journal fur die reine und ange-wandte Mathematik 2012 (2012), 189-223.

[KL10] M. Keller, D. Lenz, Unbounded Laplacians on Graphs: Ba-sic Spectral Properties and the Heat Equation, Mathematicalmodeling of natural phenomena: Spectral Problems 5 (2010),198-224.

[KLW13] M. Keller, D. Lenz, R. Wojciechowski, Volume growth, spec-trum and stochastic completeness of infinite graphs, Mathe-matische Zeitschrift 274 (2013), 905-932.

[BGK13] M. Bonnefont, S. Golenia, M. Keller, Eigenvalue asymptoticsfor Schrodinger operators on sparse graphs, arXiv:1311.7221.

[HK14] B. Hua, M. Keller, Harmonic functions of general graph Lapla-cians, Calculus of Variations and Partial Differential Equa-tions 51 (2014), 343–362.

7

8 STRUCTURE OF THE WORK

[HKMW13] X. Huang, M. Keller, J. Masamune, R. Wojciechowski, A noteon self-adjoint extensions of the Laplacian on weighted graphs,Journal of Functional Analysis 265 (2013), 1556-1578.

[BKW14] F. Bauer, M. Keller, R. Wojciechowski, Cheeger inequalitiesfor unbounded graph Laplacians, to appear in Journal of theEuropean Mathematical Society.

[HKW13] S. Haeseler, M. Keller, R. Wojciechowski, Volume growth andbounds for the essential spectrum for Dirichlet forms, Journalof the London Mathematical Society 88 (2013), 883–898.

[BHK13] F. Bauer, B. Hua, M. Keller, On the lp spectrum of Laplacianson graphs, Advances in Mathematics 248 (2013), 717-735.

[Kel10] M. Keller, The essential spectrum of the Laplacian on rapidlybranching tessellations, Mathematische Annalen 346 (2010),51-66.

[KP11] M. Keller, N. Peyerimhoff, Cheeger constants, growth andspectrum of locally tessellating planar graphs, MathematischeZeitschrift 268 (2011), 871-886.

[Kel11] M. Keller, Curvature, geometry and spectral properties of pla-nar graphs, Discrete & Computational Geometry 46 (2011),500-525.

Part 1

Introduction

Synopsis

The impact of the geometry on the spectral and stochastic featuresof Laplacians and their semigroups is studied in many areas of mathe-matics. Indeed, Laplacians on Riemannian manifolds and graphs sharea lot of common elements. Despite of this, various geometric notionssuch as distance and curvature which arise canonically in the Riemann-ian framework have no immediate analogue in the discrete setting. Fordistances it even turns out that the naive approach to define a distancevia the combinatorial graph distance leads to serious disparities in thecomparison of the theory of discrete and continuum models. The devel-opment and investigation of suitable notions of distance and curvatureis a major theme of this work.

The guiding perspective is that Laplacians on Riemannian man-ifolds and on graphs both originate from so called Dirichlet forms.From a physical perspective, such quadratic forms may intuitively beunderstood to model an ‘energy functional.’ In our presentation, we fo-cus on Dirichlet forms on discrete spaces. These spaces have the virtuethat they often allow for a very explicit and rather non-technical treat-ment. Nevertheless, our presentation is designed to pave the way for atreatment in the general case.

In the first chapter, we introduce the set up and basic concepts. Inparticular, we present a one-to-one correspondence between (weighted)graphs and regular Dirichlet forms on a discrete space. We introduceLaplacians and their semigroups via these forms, discuss their basicproperties and give examples. Next, we take a look at the heat equa-tion and a property called stochastic completeness which serves as acharacterization for uniqueness of bounded solutions to the heat equa-tion. The original treatment of these two sections is found in the orig-inal works [KL12, KL10] attached in the second part. The chapteris completed by the discussion of three classes of examples. The firstclass consists of graphs over certain measure spaces, namely, the mea-sure of vertices is assumed to be bounded below by a positive constant.The results here are based on [KL12, BHK13]. Secondly, we considergraphs with a weak form of spherical symmetry and study various spec-tral and probabilistic properties [KLW13]. Finally, we take a closerlook at sparse graphs based on results obtained in [BHK13, KL10].

11

12 SYNOPSIS

In the second chapter, we consider a notion of distance and corre-sponding notions such as volume. Here, the naive approach to definea distance for graphs by a version of the combinatorial graph distanceleads to disparities compared to Riemannian manifolds. This was firstobserved by Wojciechowski [Woj08, Woj11], see also [CdVTHT11a,KLW13]. In particular, these disparities appear when one considersunbounded operators on graphs. On the other hand, in the case of thenormalized graph Laplacian – which is always a bounded operator –the combinatorial graph distance provides the proper analogue to theRiemannian case. Therefore, the combinatorial graph distance is, insome sense, the natural metric for the normalized Laplacian while it isunsuitable for general operators. This suggests that one should look foran appropriate notion of distance for a given operator. This approachproved to be very effective in the context of strongly local regularDirichlet forms. There, so called intrinsic metrics were used to extendvarious results from Riemannian geometry to a very general framework,see the systematic pioneering investigation of such metrics by Sturm[Stu94]. Recently, a concept of intrinsic metric was introduced for gen-eral regular Dirichlet forms by Frank/Lenz/Wingert [FLW14] (whichcirculated as a preprint 5 years prior to its publication). Here, we usesuch metrics to study operators on general graphs. In this way, we ob-tain results which seem to be the natural discrete analogues to the Rie-mannian setting. As a highlight of this chapter we point out a Cheegerinequality from [BKW14]. This comes, in some sense, as a surprisesince, at first glimpse, it is not clear how a notion of distance entersthe definition of an isoperimetric constant. Indeed, this solves an openproblem of Dodziuk/Kendall [DK86] from 1986. Next to this result,Liouville theorems of Yau and Karp, Gaffney’s theorem for essentialselfadjointness, Brooks’ and Sturm’s upper bounds for the bottom ofthe (essential) spectrum and Sturm’s p-independence of the spectra areproven for graphs, see [HK14, HKMW13, HKW13, BHK13] forthe original manuscripts. All of these results are the first in this direc-tion for general graphs. They all contain the normalized Laplacian asa special case and sometimes improve the result known for this case.

In the third and final chapter, we address a notion of curvature.We restrict ourselves to planar graphs with standard weights and con-sider a very intuitive geometric notion of curvature going back to ideasof Descartes. Our main focus lies on spectral consequences of uppercurvature bounds. We give lower and upper bounds on the bottom ofthe spectrum in terms of curvature as they were obtained in [KP11].Furthermore, we characterize discreteness of the spectrum in terms ofcurvature. This is an analogue to a theorem of Donnelly/Li and isfound in [Kel10]. In this case, we can determine the first order of

SYNOPSIS 13

the eigenvalue asymptotics which is an application of [BGK13]. Fi-nally, we apply the results of [BHK13] to discuss p-independence ofthe spectrum.

Notation

We list the most important notation of this thesis to serve as asmall index. The page number refers to the place where these objectsare defined.

• X . . . a discrete infinitely countable space (page 17).• (b, c) . . . a (weighted) graph with edge weight b and killing

term c (page 17).• m . . . a measure on X (page 18).• n . . . the normalizing measure for a graph on X which is the

sum of the edge weights about a vertex and the killing term(page 18).• Deg . . . the weighted vertex degree which is the sum of the edge

weights about a vertex divided by the measure (page 36).• deg . . . the combinatorial vertex degree which is the number

of edges emanating from a vertex (page 18).• Q onD . . . the generalized form associated to a graph (page 18).• Q . . . the regular Dirichlet associated to a graph (page 21).• L on F . . . the generalized Laplacian associated to a graph

(page 19).• L . . . the selfadjoint operator associated to Q which is a re-

striction of L (page 21).• ∆ . . . the Laplacian on a graph with standard weights and the

counting measure (page 23).• ∆n . . . the normalized Laplacian on a graph with standard

weights and the normalizing measure (page 23).

15

CHAPTER 1

Dirichlet forms on discrete spaces

This chapter is dedicated to setting the stage and introducing thebasic notions and concepts. We start by defining weighted graphs on adiscrete measure space and show that there is a one-to-one correspon-dence to regular Dirichlet forms on this space. Via these forms, weobtain selfadjoint operators on the corresponding `2 space by generaltheory and characterize basic features such as boundedness and the factthat compactly supported functions are in the domain of these opera-tors. These operators give rise to a semigroup which can be extendedto the `p spaces for all 1 ≤ p ≤ ∞. We discuss that their generatorsare restrictions of a general Laplacian.

Secondly, we discuss the heat equation for bounded functions. Unique-ness of solutions can be characterized by a property called stochasticcompleteness at infinity. Furthermore, the stability of this propertyunder embeddings into supergraphs is discussed.

Finally, we discuss classes of examples of graphs which allow for amore specific investigation of certain aspects. First, we consider graphsover a measure space whose measure allows for a positive lower boundon singelton sets. Secondly, we study graphs with a weak sphericallysymmetry and, thirdly, we look at graphs with relatively few edgeswhich we refer to as sparse graphs.

The first three sections summarize the results of the original man-uscript [KL12] and also some of the material in [KL10, BHK13].The first section also appeared as an introductory section in the surveyarticle [Kel14b]. The fourth section presents the results of [KLW13]and the fifth section is mainly taken from [BGK13] and from [KL10]in one instance.

1.1. Graphs and Laplacians

1.1.1. Graphs. Let X be a discrete and countably infinite space.A graph (b, c) over X is a symmetric function b : X×X → [0,∞) withzero diagonal and

∑

y∈Xb(x, y) <∞, x ∈ X,

and c : X → [0,∞). We say two vertices x, y ∈ X are neighbors ifb(x, y) > 0 and we write x ∼ y. We can think of b(x, y) as the bondstrength which increases with the strength of the interaction between

17

18 1. DIRICHLET FORMS ON DISCRETE SPACES

x and y. The function c can be thought to describe one-way-edges to avirtual point at infinity or as a potential or as a killing term. If c ≡ 0,then we speak of b as a graph over X.

We call a sequence (xk) of pairwise distinct vertices a path if xk ∼xk+1 for all indices k. We say a graph is connected if for all x, y ∈ Xthere is a finite path with x and y as end vertices. If a graph is notconnected we may restrict our attention to the connected components.Therefore, we assume in the following that the graphs under consider-ation are connected.

A measure of full support on X is given by a function m : X →(0,∞) which is extended additively to sets via m(A) =

∑x∈Am(x),

A ⊆ X. If we fix a measure m, then we speak of a graph (b, c) or bover (X,m).

Given a graph (b, c), an important special case of a measure is thenormalizing measure n defined as

n(x) =∑

y∈Xb(x, y) + c(x), x ∈ X.

Another important special case is the counting measure m ≡ 1.We say a graph is locally finite if every vertex has only finitely

many neighbors, that is, if the combinatorial vertex degree deg is finiteat every vertex:

deg(x) = #y ∈ X | x ∼ y <∞, x ∈ X.

We say a graph has standard weights if b : X × X → 0, 1 andc ≡ 0. In this case, the normalizing measure n equals the combinatorialvertex degree deg. Obviously, by the summability assumption on b,graphs with standard weights are locally finite.

1.1.2. General forms and Laplacians. In this subsection, weintroduce the forms and Laplacians on rather large spaces. Later, werestrict these objects to spaces with rather more structure.

We let C(X) be the set of real valued functions on X and Cc(X)be the subspace of functions in C(X) of finite support.

1.1.2.1. The general form. For a graph (b, c) over X, the generalquadratic form Q : C(X)→ [0,∞] is given by

Q(f) =1

2

∑

x,y∈Xb(x, y)|f(x)− f(y)|2 +

∑

x∈Xc(x)|f(x)|2

with domain

D = f ∈ C(X) | Q(f) <∞.

1.1. GRAPHS AND LAPLACIANS 19

Since Q 12 is a semi norm and satisfies the parallelogram identity, by

polarization, Q yields a semi scalar product on D via

Q(f, g) =1

2

∑

x,y∈Xb(x, y)(f(x)− f(y))(g(x)− g(y)) +

∑

x∈Xc(x)f(x)g(x).

1.1.2.2. The general Laplacian and Green’s formula. For functionsin

F = f ∈ C(X) |∑

y∈Xb(x, y)|f(y)|2 <∞ for all x ∈ X,

we define the general Laplacian L : F → C(X) by

Lf(x) =1

m(x)

∑

y∈Xb(x, y)(f(x)− f(y)) +

c(x)

m(x)f(x).

Obviously, we have F = C(X) in the locally finite case and, in general,Cc(X) ⊂ F .

Next, we come to a Green’s formula which was first shown in[HK11], see also [HKLW12].

Lemma 1.1 (Green’s formula, Lemma 4.7 in [HK11]). For f ∈ Fand ϕ ∈ Cc(X)

1

2

∑

x,y∈Xb(x, y)(f(x)− f(y))(ϕ(x)− ϕ(y)) +

∑

x∈Xc(x)f(x)ϕ(x)

=∑

x∈XLf(x)ϕ(x)m(x) =

∑

x∈Xf(x)Lϕ(x)m(x).

1.1.2.3. Solutions and harmonic functions. An important tool tostudy various analytic and probabilistic properties of graphs are so-lutions to certain equations. Here, we briefly introduce solutions toelliptic equations. Later, in Section 1.2, we consider also solutions tothe heat equation.

A function f ∈ F is called a solution (respectively, subsolution orsupersolution) for λ ∈ R if (L − λ)f = 0 (respectively, (L − λ)f ≤ 0or (L − λ)f ≥ 0). A solution (respectively, subsolution or supersolu-tion) for λ = 0 is is said to be harmonic (respectively, subharmonic orsuperharmonic).

We say a function f ∈ C(X) is positive if f(x) ≥ 0, x ∈ X, andnon-trivial and strictly positive if f(x) > 0, x ∈ X.

A Riesz space is a linear space equipped with a partial orderingwhich is compatible with addition, scalar multiplication and where themaximum and the minimum of two functions exist. Immediate exam-ples are C(X), Cc(X), F , D, and the canonical `p-spaces, 1 ≤ p ≤ ∞,introduced below.

An important fact that is needed in the subsequent material is thefollowing. In order to study existence of non-constant (respectively,


non-zero) solutions for λ ≤ 0 in a Riesz space, it suffices to studypositive subharmonic functions. The well-known lemma below followsfrom the fact that the positive and negative parts and the modulus ofa solution to λ ≤ 0 are non-negative subharmonic functions.

Lemma 1.2. Let (b, c) be a connected graph over (X,m) and F0 ⊆ Fbe a Riesz space. If there are no non-constant positive subharmonicfunctions in F0, then there are no non-constant solutions for λ ≤ 0 inF0 and, in particular, any solution for λ < 0 is zero.

1.1.3. Dirichlet forms and their generators. The forms andLaplacians introduced above are defined on spaces without a norm.Next, we will consider restrictions of these objects to Hilbert and Ba-nach spaces.

Let `p(X,m) be the canonical real valued `p-spaces, p ∈ [1,∞], withnorms

‖f‖p =(∑

x∈X|f(x)|pm(x)

) 1p, p ∈ [1,∞),

‖f‖∞ = supx∈X|f(x)|.

As `∞(X,m) does not depend on m, we also write `∞(X). For p = 2,we have a Hilbert space `2(X,m) with scalar product

〈f, g〉 =∑

x∈Xf(x)g(x)m(x), f, g ∈ `2(X,m),

and we denote the norm by ‖ · ‖ = ‖ · ‖2.For the domain of the general Laplacian F , one always has

`∞(X) ⊆ F .The inclusion of `p(X,m), p ∈ [1,∞), in F does not hold in general,but holds in the case of uniformly positive measure, that is,

infx∈X

m(x) > 0.

1.1.3.1. Dirichlet forms. In our context, a Dirichlet form is a closedquadratic form q with domain D ⊆ `2(X,m) such that, for f ∈ D, wehave 0 ∨ f ∧ 1 ∈ D and

q(0 ∨ f ∧ 1) ≤ q(f),

where g ∨ h denotes the maximum of g and h and g ∧ h denotes theminimum of g and h. A form q is called regular if D ∩ Cc(X) is dense

in D with respect to ‖ · ‖q = (q(·) + ‖ · ‖2)12 and in Cc(X) with respect

to ‖ · ‖∞.We denote the restriction of Q to

D(Q(N)) = D ∩ `2(X,m)

1.1. GRAPHS AND LAPLACIANS 21

by Q(N), where the superscript (N) indicates “Neumann boundaryconditions.” By Fatou’s lemma, Q(N) can be seen to be lower semi-continuous and, thus, closed. It follows directly that Q(N) is a Dirichletform.

Moreover, closedness of Q(N) yields immediately that the restrictionof Q to Cc(X) is closable. We define Q = Q(D) by

D(Q) = Cc(X)‖·‖Q

,

Q(f) =1

2

∑

x,y∈Xb(x, y)|f(x)− f(y)|2 +

∑

x∈Xc(x)|f(x)|2, f ∈ D(Q).

Here, the superscript (D) indicates “Dirichlet boundary conditions.”It can be seen that Q is a Dirichlet form (see [FOT11, Theorem 3.1.1]for a proof in the general setting and, for a proof of this fact in thegraph setting, see [Sch12, Proposition 2.10]). Obviously, Q is regular.

As it turns out, by [KL12, Theorem 7], all regular Dirichlet formson (X,m) are given in this way – a fact which can be also derived fromthe Beurling-Deny representation formula [FOT11, Theorem 3.2.1 andTheorem 5.2.1].

Theorem 1.3 (Theorem 7 in [KL12]). If q is a regular Dirichletform on `2(X,m), then there is a graph (b, c) such that q = Q(D).

1.1.3.2. Markovian semigroups and their generators. By general the-ory (see e.g. [Wei80, Satz 4.14]), Q yields a positive selfadjoint oper-ator L with domain D(L) viz

Q(f, g) = 〈L 12f, L

12 g〉, f, g ∈ D(Q).

By the second Beurling-Deny criterion, L gives rise to a Markoviansemigroup e−tL, t > 0, which extends consistently to all `p(X,m),p ∈ [1,∞], and is strongly continuous for p ∈ [1,∞). Markovian meansthat for functions 0 ≤ f ≤ 1, one has 0 ≤ e−tLf ≤ 1.

We denote the generators of e−tL on `p(X,m), p ∈ [1,∞), by Lp,that is,

D(Lp) =f ∈ `p(X,m) | g = lim

t→0

1

t(I − e−tL)f exists in `p(X,m)

Lpf = g

and L∞ is defined as the adjoint of L1. It is a direct consequence fromthe Green’s formula that L2 is a restriction of L. Moreover, in [KL12],it is also shown that Lp, p ∈ [1,∞], are restrictions of L.

Theorem 1.4 (Theorem 5 in [KL12]). Let (b, c) be a graph over(X,m) and p ∈ [1,∞]. Then,

Lpf = Lf, f ∈ D(Lp).


1.1.3.3. Boundedness of the operators. We next comment on theboundedness of the form Q and the operator L. The theorem below istaken from [HKLW12] and an earlier version can be found in [KL10,Theorem 11].

Theorem 1.5 (Theorem 9.3 in [HKLW12]). Let (b, c) be a graphover (X,m). Then the following are equivalent:

(i) X → [0,∞), x 7→ 1m(x)

(∑y∈X b(x, y) + c(x)

)is a bounded

function.(ii) Q and, in particular Q, is bounded on `2(X,m).

(iii) L and, in particular Lp, is bounded for some p ∈ [1,∞].(iv) L and, in particular Lp, is bounded for all p ∈ [1,∞].

Specifically, if the function in (i) is bounded by D < ∞, then Q ≤ 2Dand ‖Lp‖ ≤ 2D, p ∈ [1,∞].

1.1.3.4. The compactly supported functions as a core. It shall beobserved that Cc(X) is, in general, not included in D(L). Indeed,one can give a characterization of this situation. The proof is ratherimmediate and we refer to [KL12, Proposition 3.3] or [GKS12, Lemma2.7] for a reference.

Lemma 1.6. Let (b, c) be a graph over (X,m). Then the followingare equivalent:

(i) Cc(X) ⊆ D(L).(ii) LCc(X) ⊆ `2(X,m)

(iii) For each x ∈ X, the function X → [0,∞), y 7→ 1m(y)

b(x, y) is

in `2(X,m).

In particular, the above assumptions are satisfied if the graph is locallyfinite or

infy∼x

m(y) > 0, x ∈ X.

Moreover, either of the above assumptions implies `2(X,m) ⊆ F .

Proof. The equivalence of (ii) and (iii) follows from the abstractdefinition of the domain of L. The equivalence of (i) and (ii) is adirect calculation, see [KL12, Proposition 3.3] and the “in particular”statements are also immediate, see [KL12, GKS12].

1.1.3.5. Graphs with standard weights. In this subsection, we con-sider two important special cases. We say a graph has standard weightsif b : X × X → 0, 1 and c ≡ 0. For these graphs, we consider twomeasures which play a prominent role in the literature.

1.2. THE HEAT EQUATION 23

For the counting measure m ≡ 1, we denote the arising operator Lon `2(X) = `2(X, 1) by ∆ . It operates as

∆f(x) =∑

y∼x(f(x)− f(y)), f ∈ D(∆), x ∈ X.

By Lemma 1.5, the operator ∆ is bounded if and only if deg is bounded.By Lemma 1.6, we still have Cc(X) ⊆ D(∆) in the unbounded casesince standard weights imply local finiteness.

For the normalizing measure n = deg, we call the arising operatorL on `2(X, deg) the normalized Laplacian and denote it by ∆n. Theoperator ∆n acts as

∆nf(x) =1

deg(x)

∑

y∼x(f(x)− f(y)), f ∈ `2(X, deg), x ∈ X,

and, by Lemma 1.5, the operator ∆n is bounded by 2.

1.2. The heat equation

In this section, we discuss the heat equation on `∞(X). A functionu : [0,∞) → `∞(X), t 7→ ut that is continuous on [0,∞) and differen-tiable on (0,∞) at every x ∈ X is called a solution to the heat equationwith initial condition f ∈ `∞(X) if

−Lut = ∂tut, t > 0,

u0 = f.

Continuity for t 7→ Lut(x), x ∈ X, on [0,∞) can easily be seen andthe validity of the heat equation extends to t = 0.

1.2.1. Stochastic completeness. In the case c ≡ 0, uniquenessof solutions to heat equation in `∞(X) can be characterized by a prop-erty that is called stochastic completeness (or conservativeness or hon-esty or non-explosion, depending on the context). This property dealswith the question of whether the semigroup leaves the constant func-tion 1 invariant.

There is a huge body of literature from various mathematical fieldsinvestigating this property, so for references, we restrict ourselves tomentioning the work for discrete Markov processes in the late 50’sby Feller [Fel58, Fel57] and Reuter [Reu57], for manifolds the workof Azencott [Aze74] and of Grigor’yan [Gri88, Gri99], for positivecontraction semigroups the work of Arlotti/Banasiak [AB04] and ofMokhtar-Kharroubi/Voigt [MKV10] and, for strongly local Dirichletforms, the work of Sturm [Stu94].

A graph is called stochastically complete if

e−tL1 = 1

for some (all) t > 0. There is a physical interpretation of this property.This concerns the question of whether heat leaves the graph in finite


time. Assume the graph is not stochastically complete, i.e., e−tL1 < 1for some t > 0. Let 0 ≤ f ∈ `1(X,m) model the distribution of heat inthe graph at time t = 0. Then, the distribution of heat at time t > 0is given by e−tLf and the amount of heat at time t > 0 is given by∑

x∈Xe−tLf(x)m(x) = 〈e−tLf, 1〉 = 〈f, e−tL1〉 < 〈f, 1〉 =

∑

x∈Xf(x)m(x),

where the right hand side is the amount of heat at time t = 0. Hence,the amount of heat in the graph decreases in time, if the graph is notstochastically complete.

1.2.2. Stochastic completeness at infinity. Usually, stochas-tic completeness is studied for forms with vanishing killing term sincea non-vanishing killing term immediately results in the loss of heat.Here, we consider all regular Dirichlet forms on (X,m), including non-vanishing killing term and study a property called stochastic complete-ness at infinity. So, to deal with a non-vanishing c, we have to replacee−tL1 by the function

Mt(x) = e−tL1(x) +

∫ t

0

(e−sL

c

m

)(x)ds, x ∈ X.

In [KL12] it is shown that Mt is well defined, satisfies 0 ≤ Mt ≤ 1,and, for each x ∈ V , the function t 7→ Mt(x) is continuous and evendifferentiable.

In the special case c ≡ 0, we obtain Mt = e−tL1, whereas for c 6≡ 0we obtain Mt > e−tL1 for connected graphs.

By the interpretation above, the term e−tL1 can be seen to be theamount of heat contained within the graph at time t and the integralcan be interpreted as the amount of heat killed within the graph upto the time t by the killing term. Thus, Mt can be interpreted as theamount of heat, which has not been transported to the “boundary” ofthe graph at time t.

The following theorem is one of the main results of [KL12]. Forrelated results in the case c ≡ 0 and m ≡ 1 see [Fel58, Fel57, Reu57,Woj08].

Theorem 1.7. (Theorem 1 in [KL12]) Let (b, c) be a graph over(X,m). Then, for any λ < 0, the function

w :=

∫ ∞

0

−λetλ(1−Mt)dt

satisfies 0 ≤ w ≤ 1, solves (L−λ)w = 0, and is the largest non-negativel ≤ 1 with (L − λ)l ≤ 0. In particular, the following assertions areequivalent:

(i) Mt ≡ 1 for some (all) t > 0.(ii) The function w is nontrivial.

1.2. THE HEAT EQUATION 25

(iii) For any (some) λ < 0, there is no non-trivial u ∈ `∞(X) suchthat

Lu = λu.

(iv) For any (some) f ∈ `∞(X) there is a unique solution U :[0,∞)→ `∞(X), t 7→ ut to

−Lu = ∂tu, u0 = f.

Definition 1.8. We say a graph is stochastically complete at infin-ity if one of the equivalent assertions of the theorem above is fulfilled.

A further equivalence by an weak Omori-Yau-Principle can be foundin [Hua11b] for c ≡ 0.

1.2.3. Subgraphs. Next, we discuss stochastic completeness (atinfinity) from the perspective of a graph having subgraphs with thisproperty. It is well known from the theory of random walks that a graphis transient whenever it has a transient subgraph. Such a statementis wrong for stochastic completeness and stochastic completeness atinfinity.

First, we present a result that shows that a graph can be “stochas-tically completed at infinity” by adding a killing term.

Theorem 1.9 (Theorem 2 in [KL12]). For any graph (b, c) over(X,m), there is c′ : X → [0,∞) such that (b, c + c′) is stochasticallycomplete at infinity.

The idea behind the proof given in [KL12] is that through theadditional killing term so much heat is already killed in the graph thatno heat reaches the “boundary” in finite time.

Secondly, we present a result that shows the phenomena discussedabove. Namely, we can “complete” a graph by embedding the graphinto a larger supergraph.

A subgraph (bW , cW ) of a graph (b, c) over (X,m) is given by a subsetW of X and the restriction bW of b to W ×W and the restriction cWof c to W .

The graph (b, c) is then called a supergraph to (bW , cW ). Given ameasure m on X we denote its restriction to W by mW . The subgraph(bW , cW ) then gives rise to a form QW on the closure of Cc(W ) in`2(W,mW ) with respect to ‖ · ‖Q with associated operator LW .

Theorem 1.10 (Theorem 3 in [KL12]). Any graph is the subgraphof a graph that is stochastically complete at infinity. This supergraphcan be chosen to have a vanishing killing term if the original graph hasa vanishing killing term.

The idea of the proof in [KL12] is to attach sufficiently manystochastically complete graphs to each vertex. For example, one maychoose line graphs or simply single edges.


Given the two results above, it seems desirable to give a sufficientcondition for a graph not being stochastically complete at infinity interms of subgraphs.

To this end, we introduce subgraphs with Dirichlet boundary con-ditions. Given a graph (b, c) over (X,m) and W ⊆ X, the subgraph

with Dirichlet boundary conditions (b(D)W , c

(D)W ) over (W,mW ) is given

by

b(D)W = bW and c

(D)W = cW + dW ,

where dW (x) :=∑

y∈X\W b(x, y), x ∈ W . Analogous to the definition

above, we get a form Q(D)W on `2(X,m) and an operator L

(D)W .

With this terminology we get a result that complements the theo-rem above.

Theorem 1.11 (Theorem 4 in [KL12]). A graph is not stochas-tically complete at infinity, whenever it has a subgraph with Dirichletboundary conditions that is not stochastically complete at infinity.

1.3. Uniformly positive measure

In this section, we discuss a class of graphs whose measure can bebounded by a positive constant from below. It seems that these resultshave no direct analogues in the non-discrete setting.

We first show a Liouville theorem. As a consequence, we obtaina criterion for essential selfadjointness, equality of the Dirichlet andNeumann form and an explicit description of the domain of the `p

generators. These results are taken from [KL12]. Secondly, we discussa spectral inclusion result for the spectrum σ(L2) of the `2 generatorL2 in the spectrum σ(Lp) of the `p generators Lp which is taken from[BHK13].

The condition below is on the measure space (X,m) only. We saythe measure m is uniformly positive if

(M) infx∈X m(x) > 0.

For example, this holds if m is constant as in the case of the countingmeasure or deg. For some results, we may weaken (M) to a conditionthat additionally takes into account the combinatorial structure of agraph over (X,m):

(M*)∑∞

n=1 m(xn) =∞ for all infinite paths (xn).

1.3.1. A Liouville theorem. The following theorem is a slightlystronger statement than [KL12, Lemma 3.2]. The argument of theproof is the same as in [KL12] and, as it is very simple, we include ithere.

Theorem 1.12. Let (b, c) be a connected graph over (X,m) sat-isfying (M*). Then, any positive subharmonic function in `p(X,m),p ∈ [1,∞), is zero.

1.3. UNIFORMLY POSITIVE MEASURE 27

Proof. Let f be positive and subharmonic. Then, Lf(x) ≤ 0evaluated at some x gives

f(x) ≤ 1∑y∈X b(x, y)

∑

y∈Xb(x, y)f(y)

Thus, whenever there is x′ ∼ x with f(x′) < f(x) there must bey ∼ x such that f(x) < f(y). By connectedness, such x′ and x existwhenever f is non-constant. Letting x0 = x, x1 = y and proceedinginductively, there is a sequence (xn) of vertices such that 0 ≤ f(x) <f(xn) < f(xn+1), n ≥ 0, since we assumed that f is positive. Now,(M*) implies that f is not in `p(X,m), p ∈ [1,∞). On the other hand,if f is constant, then (M*) again implies f ≡ 0.

1.3.1.1. Domain of the generators. We discuss an application of theabove theorem to determine the domain of the generator on `p. Theother essential ingredient of the proof is that Lp is a restriction of L.

Theorem 1.13 (Theorem 5 in [KL12]). Let (b, c) be a graph over(X,m) such that every infinite path has infinite measure, (M*). Then,

D(Lp) = f ∈ `p(X,m) | Lf ∈ `p(X,m) for all p ∈ [1,∞).

1.3.1.2. Uniqueness of the form and essential selfadjointness. Inthe `2 case, we get the following theorem that shows that the formwith Dirichlet and Neumann boundary conditions coincide and we geta result on essential selfadjointness.

Theorem 1.14 (Theorem 6 in [KL12]). Let (b, c) be a graph over(X,m) such that every infinite path has infinite measure (M*). Then,

Q(D) = Q(N).

If, additionally, LCc(X) ⊆ `2(X,m), then Cc(X) ⊆ D(L) and therestriction of L to Cc(X) is essentially selfadjoint.

Remark. Recall that if the graph has uniformly positive mea-sure (M), this both implies (M*) and also LCc(X) ⊆ `2(X,m) byLemma 1.6.

Let us comment on the history of essential selfadjointness results forgraph Laplacians. For standard weights and the counting measure sucha result was first shown by Wojciechowski [Woj08]. The first correctproof in the general case is found in [KL12]. Later, somewhat weakerresults were obtained by [JP11] and, independently, by [TH10]. Re-sults of this type involving magnetic Schrodinger operators were laterproven in [Gol14, GKS12].


1.3.2. Spectral inclusion. In this section we discuss the spectralinclusion σ(L2) ⊆ σ(Lp) under the assumption of uniformly positivemeasure. This result was proven in [BHK13].

Theorem 1.15 (Theorem 2 in [BHK13]). Let (b, c) be a graph over(X,m) with uniformly positive measure (M). Then, for any p ∈ [1,∞],

σ(L2) ⊆ σ(Lp).

The basic observation for the proof of the theorem is that (M)implies `p(X,m) ⊆ `q(X,m), 1 ≤ p ≤ q ≤ ∞, and, thus,

D(Lp) ⊆ D(Lq), 1 ≤ p ≤ q <∞,as under the assumption (M), we can determine D(Lp) explicitly byTheorem 1.13.

The abstract reason behind the result above is that, in the case ofuniformly positive measure, the semigroup e−tL is ultracontractive.

In Section 2.6 we show that even an equality holds under a certainvolume growth assumption (even without the assumption of uniformpositivity of the measure). In general the inclusion is strict. For ex-ample for a regular tree with standard weights, the bottom of the `2

spectrum of the normalized Laplacian is known to be positive. Onthe other hand, the normalized Laplacian is bounded and, thus, theconstant function 1 is in the domain of the `∞ generator and an eigen-function to the eigenvalue 0. Hence, the `∞ spectrum and by dualityalso the `1 spectrum contains 0 which is not in the `2 spectrum. (Ofcourse, the same argumentation is true for the Laplacian ∆ with stan-dard weights and the counting measure.)

1.4. Weakly spherically symmetric graphs

The next class we consider are graphs which have a weak sphericalsymmetry. Often, symmetry is defined via existence of certain auto-morphisms. Our notion is much weaker, namely, that the graphs allowfor an ordering into spheres such that certain curvature type quantitiesare spherically symmetric. For such graphs we show that the heat ker-nel is a spherically symmetric function, give a criterion for pure discretespectrum, bounds for the spectral gap and present a characterizationfor stochastic completeness at infinity. The results presented here areoriginally proven in [KLW13].

We fix a vertex o ∈ X which we call the root and consider spheresand balls

Sr = Sr(o) = x ∈ X | d(x, o) = r and Br = Br(o) =r⋃

i=0

Si(o)

about o of radius r and S−1 = ∅. Here, d(x, y) is the combinatorialgraph distance, that is, the minimal number of edges in a path connect-ing x and y. Define the outer and inner curvatures k± : X → [0,∞)

1.4. WEAKLY SPHERICALLY SYMMETRIC GRAPHS 29

by

k±(x) =1

m(x)

∑

y∈Sr±1

b(x, y), x ∈ Sr, r ≥ 0,

and let

v =c

m.

We refer to k± as curvatures since Ld(o, ·) = k−(·) − k+(·) is referredto as a curvature-type quantity, such as mean curvature, see [DK88,Hua11a, Web10].

We call a function f : X → R spherically symmetric if its valuesdepend only on the distance to the root o, i.e., if f(x) = g(r), x ∈ Sr(o),for some function g defined on N0.

Definition 1.16. A graph (b, c) over (X,m) is called weakly spher-ically symmetric if there is a vertex o such that k± and v are sphericallysymmetric functions.

1.4.1. Symmetry of the heat kernel. We start by discussingthe consequences of weak spherical symmetry for the associated heatkernel.

For the operator L we know, by the discreteness of the underlyingspace, that there exists a map

p : [0,∞)×X ×X → R,

which we call the heat kernel associated to L, with

e−tLf(x) =∑

y∈Xpt(x, y)f(y)m(y),

for all f ∈ `2(X,m), x ∈ X, t > 0. For a locally finite graph denotethe averaging operator A : `2(X,m)→ `2(X,m) by

Af(x) =1

m(Sr)

∑

y∈Sr

f(y)m(y).

We say that the heat kernel p is spherically symmetric if e−tL commuteswith A for all t > 0. In this case pt(o, ·) is a spherically symmetricfunction for all t > 0. Spherical symmetry of p can be characterized bythe graph being weakly spherically symmetric.

Theorem 1.17 (Theorem 1 in [KLW13]). A graph (b, c) over(X,m) is weakly spherically symmetric if and only if the heat kernelis spherically symmetric.

In [KLW13] also a heat kernel comparison theorem is proven in-volving the curvatures k±.


1.4.2. Spectral gap. In this section we discuss an estimate on thespectral gap for weakly spherically symmetric graphs. The spectral gapis given by the bottom of the spectrum σ(L) of L, that is,

λ0(L) = inf σ(L).

The spectrum is said to be purely discrete if the spectrum consists onlyof eigenvalues of finite multiplicity that have no accumulation point.

The geometric quantity we use to estimate λ0(L) from below in-volves the volume of balls as well as the measure of the boundary ofballs. For a set W ⊆ X, we define the boundary ∂W of W as the setof edges leaving W , i.e.,

∂W = (x, y) ∈ W ×X \W | x ∼ y.The map b can be considered as a measure on the edges and for setsU ⊆ X ×X we write

b(U) =∑

(x,y)∈Ub(x, y)

and, in particular,

b(∂W ) =∑

(x,y)∈∂Wb(x, y).

Theorem 1.18 (Theorem 3 in [KLW13]). Let (b, c) be a weaklyspherically symmetric graph over (X,m). If

a =∞∑

r=0

m(Br)

b(∂Br)<∞,

then

λ0(L) ≥ 1

a.

Moreover, the spectrum is purely discrete.

The proof uses an Allegretto-Piepenbrink type theorem as it wasproven in [HK11]. This theorem states that the bottom of the spec-trum λ0 can be characterized by the existence of positive supersolutionsfor λ ≤ λ0. Such positive supersolutions are presented for λ < 1/a. Toshow discreteness of spectrum we present positive supersolution out-side of balls Br. This yields that the bottom of the spectrum of therestricted operators is larger than the inverse of the sum in the theoremabove starting from r + 1. This yields that these bottoms converge toinfinity. As the restrictions are finite rank perturbations of the originaloperator, they share the same essential spectrum. Thus, the essentialspectrum of L must be empty.

In [KLW13] there is also a comparison theorem for the spectralgap involving the curvature quantities k±. The proof of the comparison

1.5. SPARSENESS 31

theorem uses the heat kernel comparison and a discrete version of a so-called theorem of Li [KLVW13] which extracts λ0 from pt by takinga limit.

1.4.3. Stochastic completeness at infinity. In this subsectionwe present a characterization of stochastic completeness at infinity forweakly spherically symmetric graphs by the divergence of a sum similarto the one above in Theorem 1.18.

We may also consider c as a measure on the vertices, i.e.,

c(W ) =∑

x∈Wc(x), W ⊆ X.

Theorem 1.19 (Theorem 5 in [KLW13]). A weakly sphericallysymmetric graph (b, c) over (X,m) is stochastically complete at infinityif and only if

∞∑

r=0

m(Br) + c(Br)

b(∂B(r))=∞.

1.5. Sparseness

In this section, we discuss a class of graphs with relatively fewedges, a property which we refer to as sparseness. The results presentedhere are based on [BGK13]. There they are proven for Schrodingeroperators on graphs with standard weights, which are Laplacians minusa potential. However, for general graphs the proofs carry over verbatim.

In [BGK13] a hierarchy of notions of sparseness is introduced.The most general notion involves the so-called (a, k)-sparse graphs.Stronger notions are almost sparse graphs and then sparse graphs.

For (a, k)-sparse graphs, the number of edges in a finite set are fewcompared to the boundary edges and the vertices of the set, where ais the “ratio” for the boundary edges and k for the vertices of the set.For almost sparse graphs a can be chosen to be arbitrary small at theexpense of larger k. For sparse graphs a can be chosen to be zero. Thatis, the number of edges are few with respect to the number of verticesonly. This is discussed in detail below.

There is a close relationship to graphs which satisfy a strong isoperi-metric inequality. These are graphs where the number of edges in afinite set are few with respect to the number of boundary edges. Infact, these are (a, k)-sparse graphs where k can be chosen to be zero.

For all (a, k)-sparse graphs one can determine the form domain,characterize discreteness of the spectrum and prove eigenvalue asymp-totics. These asymptotics are even better in the case of almost sparsegraphs. For sparse graphs and graphs which satisfy a strong isoperi-metric inequality, we then also discuss estimates for the bottom of thespectrum which are sharp in the case of regular trees.


1.5.1. Notions of sparseness. Let (b, c) be a graph over (X,m).We start with the most general notion of sparseness which includes theother notions as special cases.

A graph is called (a, k)-sparse for a, k ≥ 0, if for all finite W ⊆ X

b(W ×W ) ≤ a(b(∂W ) + c(W )

)+ km(W ).

In the case of standard weights the inequality reads as

2#EW ≤ a#∂W + k#W,

where EW are the edges with starting and ending vertex in W . Thiscase is treated in [BGK13]. There non-positive c are allowed as well.

A graph is called almost sparse if for all ε > 0 there is kε ≥ 0such that the graph is (ε, kε)-sparse. Finally, sparse graphs are suchgraphs where a can be chosen to be zero, i.e., a graph is called sparseor k-sparse if it is (0, k)-sparse. For graphs with standard weights, theassumption of k-sparseness reads as

2#EW ≤ k#W.

The well known concept of an isoperimetric inequality is a specialcase of (a, k)-sparseness. A graph is said to satisfy a strong isoperimet-ric inequality if there is α such that

n(W ) ≤ α(b(∂W ) + c(W )),

where n is the normalizing measure n(x) =∑

y b(x, y) + c(x), x ∈ X.In particular, it can be seen by direct calculation that a graph satisfiesan isoperimetric inequality with α > 0 if and only if it is (a, 0)-sparsewith a = (1− α)/α.

1.5.2. Characterization of the form domain. In this sectionwe characterize the form domain of Q to be a certain `2 space by (a, k)-sparseness of the graph. Furthermore, we characterize purely discretespectrum of L in this case and present estimates for the eigenvalueasymptotics.

Every function f on X induces a form on Cc(X) ⊆ `2(X,m) bypointwise multiplication. Given a form q on Cc(X) we write

f ≤ q on Cc(X),

if 〈ϕ, fϕ〉 ≤ q(ϕ, ϕ) for all ϕ ∈ Cc(X). This will be used below for the

function f = (1− a)n/m+ k with certain constants a and k.For a function f : X → R, we define

f∞ = supK⊆X finite

infx∈X\K

f(x).

In the case (n/m)∞ = ∞, we enumerate the vertices X = xjj≥0

such that (n/m)(xj) ≤ (n/m)(xj+1), j ≥ 0. Moreover, if L has purelydiscrete spectrum, then we enumerate the eigenvalues λj(L), j ≥ 0, ofL in increasing order and counting multiplicity.

1.5. SPARSENESS 33

Theorem 1.20 (Theorem 2.2 in [BGK13]). Let (b, c) be a graphover (X,m). Then the following are equivalent:

(i) The graph is (a, k)-sparse for some a, k ≥ 0.

(ii) For some a ∈ (0, 1) and k ≥ 0

(1− a)(n/m)− k ≤ Q ≤ (1 + a)(n/m) + k on Cc(X).

(iii) For some a ∈ (0, 1) and k ≥ 0

(1− a)(n/m)− k ≤ Q on Cc(X).

(iv) D(Q) = `2(X,n).

In this case, L has pure discrete spectrum if and only if (n/m)∞ =∞.Furthermore,

(1− a) ≤ lim infj→∞

λj(L)

(n/m)(xj)≤ lim sup

j→∞

λj(L)

(n/m)(xj)≤ (1 + a)

In [BGK13] it is discussed how the constants a, k and a, k can beestimated against each other.

The most involved step in the proof of the equivalence is the im-plication (i)⇒(ii) which uses isoperimetric techniques. The implica-tion (ii)⇒(iv) is obvious. Furthermore, the implication (iv)⇒(iii) fol-lows immediately from the closed graph theorem and the implication(iii)⇒(ii) is a consequence of Kato’s inequality and algebraic manipu-lation. The remaining statements are essentially a consequence of (ii)and the Min-Max principle.

1.5.3. Almost sparseness and eigenvalue asymptotics. Foralmost sparse graphs we get even better eigenvalue asymptotics.

Theorem 1.21 (Theorem 3.2 in [BGK13]). Let (b, c) be an almostsparse graph over (X,m). Then for every ε > 0 there is kε ≥ 0 suchthat on Cc(X)

(1− ε)(n/m)− kε ≤ Q ≤ (1 + ε)(n/m) + kε.

Then D(Q) = `2(X,n). Furthermore, L has pure discrete spectrum ifand only if (n/m)∞ =∞. Moreover, in this case

limj→∞

λj(L)

(n/m)(xj)= 1.

The only related results for graphs that we are aware of are foundin [Moh13] for the adjacency matrix on sparse finite graphs.

The proof follows from Theorem 1.20 and the explicit control of theconstants a, k in terms of a, k.


1.5.4. Sparseness and the bottom of the spectrum. For afunction f : X → [0,∞), we define

f0 = infK⊆X finite

supx∈X\K

f(x).

As sparse graphs are a special case of almost sparse graphs, we haveD(Q) = `2(X,n) and the same estimate for the eigenvalue asymptotics.Moreover, we get better estimates for the bottom of the spectrum.

Theorem 1.22 (Theorem 1.1 in [BGK13]). Let (b, c) be a k-sparsegraph over (X,m). Then, for any ε ∈ (0, 1),

(1− ε) nm− k

2

(1

ε− ε)≤ Q ≤ (1 + ε)

n

m+k

2

(1

ε− ε),

on Cc(X). Furthermore,

limj→∞

λj(L)

(n/m)(xj)= 1.

and

(n/m)0 − 2

√k

2

((n/m)0 −

k

2

)≤ λ0(L).

It can be seen that the estimate above is sharp in the case of regulartrees with standard weights.

1.5.5. Strong isoperimetry and the bottom of the spec-trum. In this subsection, we consider consequences of strong isoperi-metric inequalities. From [KL10, Proposition 14] the next theoremfollows immediately. The form inequality in the theorem below shouldbe compared to the inequality in the three theorems above.

Theorem 1.23 (Proposition 14 in [KL10]). Let (b, c) be a graphover (X,m) that satisfies a strong isoperimetric inequality with param-eter α > 0. Then

(1−√

1− α2)(n/m) ≤ Q ≤ (1 +√

1− α2)(n/m),

on Cc(X). In particular,

(1−√

1− α2)(n/m)0 ≤ λ0(L).

Again the estimate above is sharp in the case of regular trees withstandard weights.

CHAPTER 2

Intrinsic metrics

This chapter is dedicated to study the consequences of geomet-ric notions related to distance. The starting point of the investiga-tion is the realization that the combinatorial graph distance is notsuitable in the case of unbounded operators. In particular, if oneconsiders volume criteria for stochastic completeness, it was observedby Wojciechowski in his Ph.D. thesis [Woj08] that the criteria ob-tained for graphs differ significantly from corresponding results in thecase of manifolds. Further results in this direction were observed in[CdVTHT11a, KLW13, Woj11]. As remedy, so-called intrinsicmetrics can be used. This is the theme of this chapter.

For strongly local Dirichlet forms, intrinsic metrics have been shownto be very effective to study various topics, see [Stu94]. Recently, thisconcept was generalized to all regular Dirichlet forms. It was firstsystematically studied and applied effectively by Frank/Lenz/Wingertin [FLW14], (see also [MU11] for an earlier mentioning of the criterionfor certain non-local forms).

By the virtue of these metrics various results can be shown forgeneral graphs for the first time. From this perspective earlier resultsfor the normalized Laplacian appear as a special instance.

The relevance of intrinsic metrics stems from the fact that theyprovide the existence of suitable cut-off function. This allows for theapplication of methods known from manifolds and partial differentialequations. While applying these methods one faces the challenge of theabsence of a pointwise Leibniz rule and as a consequence the absenceof a chain rule. In some cases the mean value theorem serves as a firststep in the right direction, however, to obtain sharp results strongerestimates are of the essence.

In the first section of this chapter, we introduce intrinsic metricsin the context of graphs and discuss basic properties and examples.Secondly, we study Liouville theorems with respect to `p bounds inthe spirit of Yau and Karp. These theorems are used to determinethe domain of the generators on `p. Furthermore, we prove a resulton essential selfadjointness in the spirit of Gaffney. Then we turn tospectral estimates. First, a lower bound on the bottom of the spectrumis presented by means of an isoperimetric constant. This Cheeger in-equality solves a problem addressed by Dodziuk/Karp in 1986. Upperbounds by exponential volume growth rates in the sense of Brooks and

35

36 2. INTRINSIC METRICS

Sturm are discussed afterwards. Finally, we look into the question ofp-independence of the generators on `p. Such investigations have theirorigin in a question of Simon for Schrodinger operators and are foundfor manifolds in the work of Sturm.

Substantial parts of the presentation of this chapter are taken fromthe survey article [Kel14b].

2.1. Definition and basic facts

In this section, we introduce the concept of intrinsic metrics forgraphs. We compare this concept to other metrics that appear in theliterature. Furthermore, we present a Hopf Rinow theorem for pathmetrics on locally finite graphs and discuss important conditions whichprovide a suitable framework in the general case.

2.1.1. Definition. We call a symmetric map ρ : X ×X → [0,∞)with zero diagonal a pseudo metric if it satisfies the triangle inequality.In [FLW14, Definition 4.1] a definition of intrinsic metrics is given forgeneral regular Dirichlet forms. It can be seen by [FLW14, Lemma 4.7,Theorem 7.3] that the definition below coincides with that in [FLW14].

Definition 2.1. A pseudo metric ρ is called an intrinsic metricwith respect to a graph (b, c) over (X,m) if

∑

y∈Xb(x, y)ρ2(x, y) ≤ m(x), x ∈ X.

Notions similar to such metrics were introduced in the context ofgraphs or jump processes under the name adapted metrics in [Fol14a,Fol14b, GHM12, Hua11a, HS14, MU11].

2.1.2. Examples and relations to other metrics. In this sub-section, we explore the definition of intrinsic metrics through examplesand counter examples.

2.1.2.1. The degree path metric. A specific example of an intrinsicmetric was introduced by Huang [Hua11a, Definition 1.6.4] and alsoappeared in [Fol11]. Let the pseudo metric ρ0 : X × X → [0,∞) begiven by

ρ0(x, y) = infx=x0∼x1∼...∼xn=y

n∑

i=1

(Deg(xi−1) ∨Deg(xi)

)− 12, x ∈ X,

where Deg : X 7→ (0,∞) is the weighted vertex degree defined as

Deg(x) =1

m(x)

∑

y∈Xb(x, y), x ∈ X.

We call any metric that minimizes sums of weights over paths of edgesa path metric.

2.1. DEFINITION AND BASIC FACTS 37

It can be seen directly that ρ0 is an intrinsic metric for the graph(b, c) over (X,m):

∑

y∈Xb(x, y)ρ2

0(x, y) ≤∑

y∈X

b(x, y)

Deg(x) ∨Deg(y)≤∑

y∈X

b(x, y)

Deg(x)= m(x).

There is the following intuition behind the definition of ρ0. Considerthe Markov process (Xt)t≥0 associated to the semigroup e−tL via

e−tLf(x) = Ex(f(Xt)), x ∈ X,where Ex is the expected value conditioned on the process starting atx. The random walker modeled by this process jumps from a vertexx to a neighbor y with probability b(x, y)/

∑z b(x, z). Moreover, the

probability of not having left x at time t is given by

Px(Xs = x, 0 ≤ s ≤ t) = e−Deg(x)t.

Qualitatively, this indicates that the larger Deg(x), the faster the ran-dom walker leaves x. Looking at the definition of ρ0(x, y), the largerthe degree of either x or y the closer they are. Combining these twoobservations, we see that the faster the random walker jumps along anedge, the shorter the edge is with respect to ρ0. Of course, the jumpingtime along an edge connecting x to y is not symmetric and depends onwhether one jumps from x to y or from y to x as the degrees of x andy can be very different. In order to get a symmetric function, ρ0 favorsthe vertex with the larger degree and the faster jumping time.

There is an analogy to the Riemannian setting in terms of meanexit times of small balls. Consider a small ball Br of radius r on ad-dimensional Riemannian manifold, then the first order term of themean exit time of Br is r2/2d [Pin85]. On a locally finite graph for avertex x a “small” ball with respect to ρ0 can be thought to have radiusr = infy∼x ρ0(x, y)/2, namely this ball contains only the vertex itself.Now, computing the mean exit time of this ball gives 1/Deg(x) ≥ r2,where equality holds whenever Deg(x) = maxy∼x Deg(y).

2.1.2.2. The combinatorial graph distance. Next, we come to a met-ric that is often the most immediate choice when one considers metricson graphs. That is the combinatorial graph distance which is the pathmetric defined as

d(x, y) =

min #k ∈ N0 | there exist x0, . . . , xk with x = x0 ∼ . . . ∼ xk = y.The next lemma shows that the combinatorial graph distance is equiv-alent to an intrinsic metric if and only if the graph has bounded geom-etry. This fact was already observed in [FLW14, KLSW].

Lemma 2.2. Let (b, c) be a graph over (X,m). The following areequivalent:


(i) The combinatorial graph distance d is equivalent to an intrinsicmetric.

(ii) Deg is a bounded function.

Furthermore, if additionally c ≡ 0, then also the following is equivalent:

(iii) L is a bounded operator.

Proof. (i)⇒(ii): Let ρ be an intrinsic metric such that C−1ρ ≤d ≤ Cρ. Then,∑

x∈Xb(x, y) =

∑

x∈Xb(x, y)d2(x, y) ≤ C2

∑

x∈Xb(x, y)ρ2(x, y) ≤ C2m(x),

for all x ∈ X. Hence, Deg ≤ C2.(ii)⇒(i): Assume Deg ≤ C and consider the degree path metric ρ0.Then, ρ1 = ρ0 ∧ 1 is an intrinsic metric as well. Clearly, ρ1 ≤ d. Onthe other hand, by Deg ≤ C we immediately get ρ1 ≥ C−

12d.

The equivalence (ii)⇔(iii) follows from Theorem 1.5.

The theorem implies, in particular, that, in the case of the normal-izing measure m = n, the combinatorial graph distance is an intrinsicmetric as Deg = n/m = 1, in the case of c ≡ 0, and Deg ≤ n/m = 1 ingeneral.

Furthermore, for a graph with standard weights and the count-ing measure associated to the Laplacian ∆, the combinatorial graphdistance d is a multiple of an intrinsic metric if and only if the combi-natorial vertex degree deg is bounded since Deg = deg.

2.1.2.3. Comparison to the strongly local case. An important differ-ence to the case of strongly local Dirichlet forms is that, in the graphcase, there is no maximal intrinsic metric. For example, for a completeRiemannian manifold M the Riemannian distance dM is the maximalC1 metric ρM that satisfies

|∇MρM(o, ·)| ≤ 1,

for all o ∈ M , where ∇M is the Riemannian gradient. In fact, dM canbe recovered by the formula

dM(x, y) = supf(x)− f(y) | f ∈ C∞c (M), |∇Mf | ≤ 1, x, y ∈ X.Now, for discrete spaces, the maximum of two intrinsic metrics is

not necessarily an intrinsic metric. This is discussed next. Considerthe pseudo metric σ

σ(x, y) = supf(x)− f(y) | f ∈ A, x, y ∈ X,where

A =f : X → R |

∑

y∈Xb(x, y)|f(x)− f(y)|2 ≤ m(x) for all x ∈ X

.

2.1. DEFINITION AND BASIC FACTS 39

As discussed for the Riemannian case above, in the strongly local casethe analogue of σ defines the maximal intrinsic metric. But σ is, ingeneral, not even equivalent to an intrinsic metric in the graph case.

A basic example where this can be immediately seen can be foundin [FLW14, Example 6.2]. More generally, this phenomena can bechecked for arbitrary tree graphs with standard weights and countingmeasure to which the the operator ∆ is associated. In this case, σ = 1

2d

and, by the discussion above, we already know that the combinatorialgraph distance d is not equivalent to an intrinsic metric whenever ∆ isunbounded.

An abstract way to see that σ is, in general, not intrinsic is dis-cussed in [KLSW, Section 1]. Namely, the set of Lipschitz continuousfunctions Lipρ with respect to an intrinsic metric ρ is included in Aand Lipρ is closed under taking suprema. On the other hand, A is,in general, not closed under taking suprema. Hence, in general, Lipρis not equal to A. It would be interesting to know whether one cancharacterize the situation when these two spaces are different.

2.1.2.4. Another path metric. Colin de Verdiere/Torki-Hamza/Truc[CdVTHT11a] studied a path pseudo metric δ which is given as

δ(x, y) = infx=x0∼...∼xn=y

n−1∑

i=0

(m(x) ∧m(y)

b(x, y)

) 12, x, y ∈ X.

By a similar argument as in Lemma 2.2, this metric can be seen to beequivalent to the intrinsic metric ρ0 if and only if the combinatorialvertex degree deg is bounded on the graph.

2.1.3. A Hopf-Rinow theorem. We shall stress that, in general,an intrinsic metric ρ (and in particular ρ0) is not a metric and (X, ρ)might not even be locally compact. This can be seen from examples in[HKMW13, Example A.5]. However, for locally finite graphs and pathmetrics such as ρ0, the situation is much more tame. For example thetopology is discrete in this case. In [HKMW13] a Hopf-Rinow typetheorem is shown, see also [Mil11]. Recall that for a path γ = (xn) thelength with respect to a metric ρ is given by l(γ) =

∑j≥0 ρ(xj, xj+1)

and, moreover, a path γ = (xn) is called a geodesic with respect to ametric ρ if ρ(x0, xn) = l((x0, . . . , xn)) for all n.

Theorem 2.3 (Theorem A1 in [HKMW13]). Let (b, c) be a locallyfinite graph over (X,m) and let ρ be a path metric. Then, the followingare equivalent:

(i) (X, ρ) is complete as a metric space.(ii) (X, ρ) is geodesically complete, that is any infinite geodesic has

infinite length.(iii) The balls in (X, ρ) are finite.


2.1.4. Some important conditions. As discussed above, thetopology induced by an intrinsic metric can be rather wild. So, inthe general situation, one often has to make additional assumptions.Here, we present some of the most important assumptions and discusstheir implications.

We say a pseudo metric ρ admits finite balls if (iii) in the theoremabove is satisfied for ρ, i.e., if

(B) The distance balls Br(x) = y ∈ X | ρ(x, y) ≤ r are finite forall x ∈ X, r ≥ 0.

A somewhat weaker assumption is that the weighted vertex degreeis bounded on balls :

(D) The restriction of Deg to Br(x) is bounded for all x ∈ X,r ≥ 0.

Clearly, (B) implies (D). Moreover, (D) is equivalent to the fact thatL restricted to the `2 space of a distance ball is a bounded operator,confer Theorem 1.5.

The assumptions (B) and (D) can be understood as bounding ρ ina certain sense from below. Next, we come to an assumption whichmay be understood as an upper bound.

We say a pseudo metric ρ has finite jump size if

(J) The jump size s = supρ(x, y) | x, y ∈ X, x ∼ y is finite.

The assumptions (B) and (J) combined have the following conse-quence on the combinatoric structure of the graph.

Lemma 2.4. Let (b, c) be a graph over (X,m) and let ρ be an pseudometric. If ρ satisfies (B) and (J), then the graph is locally finite.

Proof. If there was a vertex with infinitely many neighbors, thenthere would be a distance ball containing all of them by finite jumpsize. However, this is impossible by (B).

2.2. Liouville theorems

The classical Liouville theorem in Rn states that if a harmonicfunction is bounded from above, then the function is constant. Here,we look into boundedness assumptions such as `p growth bounds andpresent results proven in [HK14].

In Section 1.3.1 we have already presented such a Liouville theoremunder the assumption of uniformly positive measure. Here, we addressthe question of arbitrary measures under some completeness assump-tion on the graph. Such results go back to Yau [Yau76] and Karp[Kar82] in the case of manifolds.

We first discuss the results for manifolds in the next subsection.Afterwards, we discuss the case of the normalized Laplacian for graphsand the results that have been proven for this operator. Finally, wepresent theorems of [HK14] that recover Yau’s and Karp’s results for

2.2. LIOUVILLE THEOREMS 41

general graph Laplacians using intrinsic metrics. As a consequence,this yields a sufficient criterion for recurrence.

The results of this section are used later to address questions such asessential selfadjointness and to determine the domain of the generators.

Throughout this section, keep in mind that absence of non-constantpositive subharmonic functions implies the absence of non-constantharmonic functions, see Lemma 1.2.

2.2.1. Historical remarks. Here, we discuss the results that pre-ceded [HK14] in the case of manifolds and graphs.

2.2.1.1. Manifolds. Let M be a connected Riemannian manifoldand ∆M the Laplace Beltrami operator. A twice continuously differen-tiable function f on M is called harmonic (respectively, subharmonic)if ∆Mf = 0 (respectively, ∆Mf ≤ 0). The assumption that f is twicecontinuously differentiable can be relaxed, but here we do not want toput the focus on the degree of smoothness.

In 1976 Yau [Yau76] proved that on a complete Riemannian man-ifold M any harmonic function or positive subharmonic function inLp(M) is constant. This result was later strengthened by Karp in 1982[Kar82]. Namely, any harmonic function or positive subharmonic func-tion f that satisfies

infr0>0

∫ ∞

r0

r

‖f1Br‖p

p

dr =∞,

is already constant, where 1Br is the characteristic function of the ge-odesic ball Br about some arbitrary point in the manifold. Yau’s the-orem is a direct consequence of Karp’s result.

Later in 1994 Sturm [Stu94] generalized Karp’s theorem to thesetting of strongly local Dirichlet forms, where balls are taken with re-spect to the intrinsic metric. The underlying assumption on the metricis that it generates the original topology and all balls are relativelycompact.

2.2.1.2. Graphs. For graphs b over (X,m), so far results in this di-rection were obtained for the normalizing measure m = n only. Inthis case, the operator L is bounded, see Section 1.5, and the combi-natorial graph distance d is an intrinsic metric, see Subsection 2.1.2.2.(Of course, harmonicity depends only on the graph b and not on themeasure m, but for the function to be in an `p space does depend onthe measure.)

Starting 1997 with Holopainen/Soardi [HS97], Rigoli/Salvatori/Vignati [RSV97], Masamune [Mas09], eventually in 2013 Hua/Jost[HJ13] showed that if a harmonic or positive subharmonic function fsatisfies

lim infr→∞

1

r2‖f1Br(x)‖pp <∞,


for some p ∈ (1,∞) and x ∈ X, then f must be constant. Here, theballs are taken with respect to the combinatorial graph distance. Thisdirectly implies Yau’s theorem for p ∈ (1,∞). Moreover, Hua/Jost[HJ13] also show Yau’s theorem for p = 1.

2.2.2. Yau’s and Karp’s theorem for general graphs. Wenow turn to general graphs equipped with an intrinsic metric. As in themanifold setting, we need a completeness assumption on the graph as ametric space. For graphs with the normalizing measure, completenessis guaranteed since the combinatorial graph distance always gives riseto a complete metric space. In the theorem below, we state a graphversion of Yau’s theorem for the general graphs. We assume that theweighted vertex degree is bounded on balls (D) and the jump size isfinite (J). In the case of a path metric on a local finite graph, the Hopf-Rinow theorem, Theorem 2.3, shows that metric completeness implies(D).

Theorem 2.5 (Corollary 1.2 in [HK14]). Let b be a graph over(X,m) and let ρ be an intrinsic metric with bounded degree on balls(D) and finite jump size (J). If f ∈ `p(X,m), p ∈ (1,∞), is a positivesubharmonic function then f is constant.

Let us mention that the case `1 is more subtle in the general case.In [HK14, Theorem 1.7] it was shown that for stochastically completegraphs Yau’s Liouville theorem remains true in the case p = 1. Theproof follows ideas from [Gri99]. Otherwise, there are counter exam-ples, see [HK14, Section 4].

Next, we turn to a graph version of Karp’s theorem which is themain result of [HK14].

Theorem 2.6 (Theorem 1.1 in [HK14]). Let b be a graph over(X,m) and let ρ be an intrinsic metric with bounded degree on balls(D) and finite jump size (J). If f is a positive subharmonic functionsuch that for some p ∈ (1,∞) and x ∈ X

infr0>0

∫ ∞

r0

r

‖f1Br(x)‖p

p

dr =∞,

then f is constant. Here, 1B is again the characteristic function of aset B ⊆ X.

In particular, the theorem above implies the result of Hua/Jost[HJ13]. It can even be seen that a harmonic function f satisfying

lim supr→∞

1

r2 log r‖f1Br(x)‖pp <∞,

for some p ∈ (1,∞), is constant.The proof of the above theorems uses a Caccioppoli inequality,

[HK14, Theorem 1.8]. This itself is proven by the use of suitable

2.3. DOMAIN OF THE GENERATORS 43

cut-off functions stemming from the intrinsic metrics. The part of thechain rule is then played by the inequality

fp−1(x)− fp−1(y) ≥ C(f(x) ∨ f(y))p−1(f(x)− f(y)), x, y ∈ Xwhich is taken from [HS14]. The proof of Theorem 2.6 then uses aninduction argument such as [Stu94].

2.2.3. Recurrence. As a direct consequence of Karp’s theoremwe get a sufficient criterion for recurrence of a graph. A connectedgraph b over X is called recurrent if for all measures m and some (all)x, y ∈ X, we have ∫ ∞

0

e−tL1x(y)dt =∞.

This is equivalent to absence of non-constant bounded subharmonicfunctions.

Analogous results to the criterion below in the case of manifoldsand strongly local Dirichlet forms are due to [Kar82, Theorem 3.5]and [Stu94, Theorem 3]. For graphs the result below generalizesthe results of [DK86, Theorem 2.2], [RSV97, Corollary B], [Woe00,Lemma 3.12], [Gri09, Corollary 1.4], [MUW12, Theorem 1.2].

Theorem 2.7 (Corollary 1.6 in [HK14]). Let b be a connectedgraph over (X,m) and let ρ be an intrinsic metric with bounded degreeon balls (D) and finite jump size (J). If for some x ∈ X

∫ ∞

1

r

m(Br(x))dr =∞,

then the graph is recurrent.

2.3. Domain of the generators and essential selfadjointness

In this section we address the question of identifying the domain ofthe generators Lp. Classically, the special case p = 2 received particularattention. Going back to investigations of Friedrichs and von Neumann,a classical question is whether a symmetric operator on a Hilbert spacehas a unique selfadjoint extension. This property is often studied underthe name essential selfadjointness.

The connection of essential selfadjointness with metric completenessis that if there exists a boundary, one might have to impose certain“boundary conditions” in order to obtain a selfadjoint operator.

We first discuss the manifold case which is often referred to asGaffney’s theorem. Secondly, we consider graphs and recover Gaffney’stheorem with the use of intrinsic metrics. Furthermore, we determinethe domain of the generators Lp on `p. The results of this section arefound in [HKMW13] and [HK14].


2.3.1. Historical remarks. Again, we discuss some of the resultsthat preceded [HKMW13] and [HK14] in the case of manifolds andgraphs.

2.3.1.1. Manifolds. A result going back to the work of Gaffney[Gaf51, Gaf54] states that, on a geodesically complete manifold, theso-called Gaffney Laplacian is essentially selfadjoint. This is equivalentto the uniqueness of the Markovian extension of the minimal Laplacian.Independently, essential selfadjointness of the Laplace Beltrami opera-tor on the compactly supported, infinitely often differentiable functionswas shown by Roelcke [Roe60]. For later results in this direction seealso [Che73, Str83].

2.3.1.2. Graphs. The first results connecting metric completenessand essential selfadjointness were obtained by Torki-Hamza [TH10],Colin de Verdiere/ Torki-Hamza/Truc [CdVTHT11a, CdVTHT11b]and Milatovic [Mil11, Mil12]. These results were proven for (mag-netic) Schrodinger operators on graphs with bounded combinatorialvertex degree and the metric δ discussed in Section 2.1.2.4. As dis-cussed there, δ is equivalent to an intrinsic metric if and only if thecombinatorial vertex degree is bounded.

2.3.2. Gaffney’s theorem for graphs. For the general case ofunbounded combinatorial vertex degree, we consider intrinsic metricsto recover a Gaffney theorem.

In Lemma 1.6 we demonstrated that we may not have LCc(X) ⊆`2(X,m) for general graphs. Hence, in general, L is not a symmetricoperator on Cc(X) as a subspace of `2(X,m). Nevertheless, one can stilldetermine whether the forms with Dirichlet and Neumann boundaryconditions are equal. Recall that we refer to the restriction of Q tothe closure of Cc(X) as the form with Dirichlet boundary conditionsQ = Q(D) and to the restriction of Q to D∩ `2(X,m) as the form withNeumann boundary conditions Q(N). Moreover, in the case of equality,we can even identify the domain of the generator.

The following result is found in [HKMW13] for graph Laplaciansand in [GKS12] for magnetic Schrodinger operators.

Theorem 2.8 (Theorem 1 in [HKMW13]). Let b be a graph over(X,m) and let ρ be an intrinsic metric with bounded degree on balls(D) and finite jump size (J). Then,

Q(D) = Q(N)

and

D(L) = f ∈ `2(X,m) | Lf ∈ `2(X,m).Furthermore, if LCc(X) ⊆ `2(X,m), then L|Cc(X) is essentially selfad-joint on `2(X,m).

2.4. ISOPERIMETRIC CONSTANTS AND LOWER SPECTRAL BOUNDS 45

Here, the assumptions (D) and (J) serve again as an analogue forthe completeness assumption. By the virtue of the Hopf Rinow typetheorem, Theorem 2.3, we immediately get the following analogue tothe classical Gaffney theorem from Riemannian geometry.

Corollary 2.9 (Theorem 2 in [HKMW13]). Let b be a locally fi-nite graph over (X,m) and let ρ be an intrinsic path metric. If (X, ρ) ismetrically complete, then L|Cc(X) is essentially selfadjoint on `2(X,m).

We can also determine the domain of the generators of the semi-group on `p.

Theorem 2.10 (Corollary 1.4 in [HK14]). Let b be a graph over(X,m) and let ρ be an intrinsic metric with bounded degree on balls(D) and finite jump size (J). Then,

D(Lp) = f ∈ `p(X,m) | Lf ∈ `p(X,m), for all p ∈ (1,∞).

Furthermore, in [HKMW13], the case of metrically incompletegraphs is treated. For locally finite graphs the capacity of the Cauchyboundary is defined. Whenever the boundary has finite capacity equal-ity of the form with Dirichlet and Neumann boundary conditions canbe characterized by the boundary having zero capacity [HKMW13,Theorem 3]. It is also shown that if the upper Minkowski codimensionof the boundary is larger than 2, then the boundary has zero capacity[HKMW13, Theorem 4].

2.4. Isoperimetric constants and lower spectral bounds

We now turn to the spectral theory of the operator L. In thissection we aim for lower bounds on the bottom of the spectrum

λ0(L) = inf σ(L)

via so called isoperimetric estimates. Such lower bounds are oftenreferred to as Cheeger’s inequality.

We first discuss the result on manifolds going back to Cheeger from1960. Then, we discuss how an analogous result was proven in the 80’sfor the normalized Laplacian by Dodziuk/Kendall and what kind ofproblems occur for the operator ∆. Finally, we examine how intrinsicmetrics can be used to overcome these problems and establish thisinequality for general graph Laplacians which is proven in [BKW14].

2.4.1. Historical remarks on Cheeger’s inequality.2.4.1.1. Manifolds. For a non-compact Riemannian manifoldM the

isoperimetric constant or Cheeger constant is defined as

hM = infS

Area(∂S)

vol(int(S)),


where S runs over all hypersurfaces cutting M into a precompact pieceint(S) and an unbounded piece. Denote by λ0(∆M) the bottom of thespectrum of the Laplace-Beltrami operator. The well known Cheegerinequality reads as

λ0(∆M) ≥ h2M

4.

See [Che70] for Cheeger’s original work on the compact case and[Bro93] for a discussion of the non-compact case.

2.4.1.2. Graphs with standard weights. There is an enormous amountof literature on isoperimetric inequalities, especially for finite graphs,see [AM85] as one of the first papers for finite graphs. Here, we restrictourselves to infinite graphs although the methods can also be appliedto the finite graph case as well.

Recall that the boundary of a set W ⊆ X is defined as the set ofedges emanating from W , i.e.,

∂W = (x, y) ∈ W ×X \W | x ∼ y.In 1984 Dodziuk [Dod84] considered graphs with standard weights andthe counting measure. The isoperimetric constant he studied is closelyrelated to

h1 = infW ⊆ X finite

|∂W ||W |

and Dodziuk’s proof yields

λ0(∆) ≥ h21

2D,

with D = supx∈X deg(x). This analogue of Cheeger’s inequality is effec-tive for graphs with bounded vertex degree. However, for unboundedvertex degree, the bound becomes trivial.

Two years later Dodziuk and Kendall [DK86] proposed a solutionto this issue by considering graphs with standard weights and the nor-malizing measure n = deg instead. The corresponding isoperimetricconstant is

hn = infW ⊆ X finite

|∂W |deg(W )

and they prove in [DK86] for the normalized Laplacian ∆n that

λ0(∆n) ≥ h2n

2.

This analogue of Cheeger’s inequality does not have the disadvantageof becoming trivial for unbounded vertex degree. This seems to be thereason that in the following the operator ∆ was rather neglected inspectral geometry of graphs and the normalized Laplacian ∆n gainedmomentum.

2.4. ISOPERIMETRIC CONSTANTS AND LOWER SPECTRAL BOUNDS 47

2.4.2. Cheeger’s inequality for graphs. The considerations inthe previous sections suggest that intrinsic metrics allow results forgeneral graph Laplacians. However, it is not obvious how isoperimetricconstants can be related to a specific metric. So, the crucial new ele-ment is to see how a metric is already hidden in the previous definitionof isoperimetric constants which worked for the normalized Laplacian.Revisiting the definition of the area of the boundary above, we findthat

b(∂W ) =∑

(x,y)∈∂Wb(x, y) =

∑

(x,y)∈∂Wb(x, y)d(x, y)

with the combinatorial graph distance d on the right hand side. Re-member that d is an intrinsic metric for the graph b over (X,n).

The new idea is to replace d by an intrinsic metric ρ for a graph bover (X,m). We define

Area(∂W ) =∑

(x,y)∈∂Wb(x, y)ρ(x, y).

That is, we take the length of an edge into consideration to measurethe area of the boundary. We define

h = infW ⊆ X finite

Area(∂W )

m(W ),

and obtain the following theorem which is found in [BKW14].

Theorem 2.11 (Theorem 1 in [BKW14]). Let b be a graph over(X,m) and let ρ be an intrinsic metric. Then,

λ0(L) ≥ h2

2.

The interesting new part of the theorem is the definition of theisoperimetric constant. Having this definition the usual proof schemeapplies which is sketched below.

Idea of the proof. The proof of the theorem is based on an areaand a co-area formula. For f ≥ 0, let

Ωt = x ∈ X | f(x) > t.Then, one can prove, using Fubini’s theorem for f ∈ Cc(X),

m(Ωt) =∑

x∈Xf(x)m(x),

Area(∂Ωt) =∑

x,y∈Xb(x, y)ρ(x, y)|f(x)− f(y)|.

The rest of the proof is basically the Cauchy-Schwarz inequality andvarious algebraic manipulations.


One may also consider potentials c ≥ 0 in the estimate by intro-ducing edges from vertices x with c(x) > 0 to virtual sibling vertices xwith edge weight b(x, x) = c(x). The union of vertices x ∈ X and x isdenoted by X. Furthermore, we extend an intrinsic metric ρ on X tothe new edges via

ρ(x, x) =(m(x)−∑y∈X b(x, y)ρ(x, y)2)

12

c(x).

The extension of ρ becomes an intrinsic metric if one chooses m(x) =m(x). Now, we define h by taking the infimum of the quotient with theextension of b and ρ as above only over subsets of X, see [BKW14,Section 5].

2.5. Volume growth and upper spectral bounds

In this section, we discuss upper bounds for the bottom of theessential spectrum

λess0 (L) = inf σess(L).

The essential spectrum σess(L) of an operator is the part of the spec-trum which does not contain discrete eigenvalues of finite multiplicity.Clearly, λ0(L) ≤ λess

0 (L).We first discuss the classical result on Riemannian manifolds going

back to Brooks and Sturm. There are corresponding results for thenormalized Laplacian. Next, we show how such a result fails in thecase of the Laplacian with respect to the counting measure based onexamples developed in [KLW13]. Finally, we employ intrinsic metricsto recover Brooks’ result for general graph Laplacians based on resultsof [HKW13]. Let us remark that the results in [HKW13] are provenin the general context of regular Dirichlet forms.

2.5.1. Historical remarks.2.5.1.1. Manifolds. Let M be a complete connected non-compact

Riemannian manifold with infinite volume. Let λess0 (∆M) be the bottom

of the essential spectrum of the Laplace Beltrami operator ∆M . LetµM be the upper exponential growth rate of the distance balls

µM = lim supr→∞

1

rlog vol(Br(x)),

for an arbitrary x ∈M . Brooks showed in 1981 [Bro81] that

λess0 (∆M) ≤ µ2

M

4.

Later, in 1996, Sturm [Stu94] showed using the lower exponentialgrowth rate of the distance balls with variable center

µM

= lim infr→∞

infx∈M

1

rlog vol(Br(x))

2.5. VOLUME GROWTH AND UPPER SPECTRAL BOUNDS 49

the following bound

λ0(∆M) ≤µ2M

4.

Indeed, the result in [Stu94] is shown in the general context of stronglylocal regular Dirichlet forms.

An immediate corollary is that for M with subexponential growth,i.e., µ

M= 0, the value 0 is in the spectrum of the Laplace Beltrami

operator.

2.5.1.2. Graphs. For graphs with standard weights and the nor-malizing measure Dodziuk/Karp [DK88] proved in 1987 the first ana-logue of Brooks’ theorem for graphs. This result was later improvedby Ohno/Urakawa [OU94] and Fujiwara [Fuj96a] resulting in the es-timate

λess0 (∆n) ≤ 1− 2eµn/2

eµn + 1

with

µn = lim supr→∞

1

rlog n(Br(x)),

for arbitrary x ∈ X and n = deg. It can be checked that the boundabove is smaller than µ2

n/8.Next, we discuss how, for graphs with standard weights and the

counting measure, such a bound fails when volume growth is consideredvia the combinatorial graph distance.

The examples are so called anti-trees which were studied by Woj-ciechowski [Woj09] as counter examples for volume bounds for stochas-tic completeness. Specifically, anti-trees are highly connected graphs.They can be characterized as follows: A vertex in a sphere (with re-spect to a root vertex) is connected to every neighbor in the succeedingsphere, (where spheres are considered with respect to the combinatorialgraph distance). See Figure 1 below for an example.

Figure 1. An anti-tree with sr+1 = 2r


For an anti-tree, let sr be the number of vertices with combinatorialgraph distance r to a root vertex. Denote, furthermore, vr = s0 + . . .+sr, r ≥ 0. In [KLW13] (confer Theorem 1.18) it was shown that

a =( ∞∑

r=0

vrsrsr+1

)−1

is a lower bound on the bottom of the spectrum λ0(∆) of ∆ (wherea = 0 if the sum diverges). Moreover, in the case where the sumconverges, the spectrum of ∆ is purely discrete, i.e., there is no essentialspectrum . In particular, this result implies that anti-trees with

sr ∼ r2+ε, ε > 0,

have positive bottom of the spectrum and purely discrete spectrum,see [KLW13, Section 6]. However, for sr ∼ r2+ε, we have vr ∼ r3+ε,that is, these are graphs of little more than cubic growth with positivebottom of the spectrum and no essential spectrum. Hence, there isno analogue to Brooks’ or Sturm’s theorem for ∆ with respect to thecombinatorial graph distance.

2.5.2. Brooks’ theorem for graphs. Let b be a graph over (X,m)and let ρ be an intrinsic metric. Let Br(x) be the distance r ball abouta vertex x with respect to the metric ρ. We define

µ = lim infr→∞

1

rlogm(Br(x)),

for fixed x ∈ X and

µ = lim infr→∞

infx∈X

1

rlogm(Br(x)).

In [HKW13] analogues of Brooks’ and Sturm’s theorem are provenfor regular Dirichlet forms. As a special case the following theorem isobtained for graphs. Folz [Fol14b] independently proved, by differentmethods, a special case of the theorem below for locally finite graphswith uniformly positive measure.

Theorem 2.12 (Corollary 4.2 in [HKW13]). Let b be a connectedgraph over (X,m) and let ρ be an intrinsic metric such that the ballsare finite (B). Then,

λ0(L) ≤µ2

8.

If furthermore m(X) =∞, then

λess0 (L) ≤ µ2

8.

The idea of the proof combines ideas of [Stu94] and a Perrson-typetheorem.

2.6. VOLUME GROWTH AND `p-INDEPENDENCE OF THE SPECTRUM 51

Idea of the proof. Let µ = lim supr→∞1r

logm(Br(x)). Then,

the functions fa = e−aρ(o,·) for a > µ/2 and fixed o ∈ X are in `2(X,m).Moreover, by the mean value theorem and the intrinsic metric propertywe find that

Q(fa) ≤a2

2

∑

x∈X|fa(x)|2

∑

y∈Xb(x, y)ρ(x, y)2 ≤ a2

2‖fa‖2

To pass from µ to µ or µ we consider

ga,r = (e2arfa − 1) ∨ 0.

Note that ga,r is supported on B2r and, therefore, ga,r is in Cc(X)whenever (B) applies. Finally, to see the statement for the essentialspectrum, we need to modify ga,r such that we obtain a sequence offunctions that converge weakly to zero. We achieve this by cutting offga,r at 1 on Br, i.e.,

ha,r = 1 ∧ ga,r.The weak convergence of ha,r to zero is ensured by the assumptionm(X) = ∞. Now, the statement follows by a Persson-type theorem[HKW13, Proposition 2.1].

We end this section with a few remarks.

Remark. (a) In [HKW13] it is also shown that the assumption(B) can be replaced by the assumption (M*) that any infinite path hasinfinite measure from Section 1.3.

(b) As a corollary, we get under the assumptions of the theorem2h ≤ µ for the Cheeger constant h defined in Section 2.4.2.

(c) By comparing the degree path metric ρ0 with the combinatorialgraph distance d on anti-trees one finds that for sr ∼ r2−ε the balls withrespect to ρ0 grow polynomially, for sr ∼ r2 they grow exponentiallyand for sr ∼ r2+ε the graph has finite diameter with respect to ρ0. Thisshows that the examples in the section above are indeed sharp.

2.6. Volume growth and `p-independence of the spectrum

In this section we turn to the spectra of the operators Lp on `p,p ∈ [1,∞] which were defined in Section 1.1.3.2. In the beginning ofthe 80’s Simon [Sim82] asked the famous question whether the spectraof certain Schrodinger operators on Rd are independent on which Lp

space they are considered. Hempel/Voigt [HV86, HV87] gave anaffirmative answer in 1986. Here, we consider a geometric analogue ofthis question. This goes back to a theorem of Sturm on Riemannianmanifolds, [Stu94]. Here, we deal with graphs. With an intrinsicmetric at hand, a corresponding result was obtained in [BHK13] whichis discussed afterwards.


2.6.1. Historical remarks. In 1993, Sturm [Stu94] proved a the-orem for uniformly elliptic operators on a complete Riemannian man-ifold M whose Ricci curvature is bounded below. We assume that Mhas uniform subexponential growth, i.e., for any ε > 0 there is C > 0such that for all r > 0 and all x ∈M

vol(Br(x)) ≤ Ceεrvol(B1(x)).

Then the spectrum of a uniformly elliptic operator on such a manifold isindependent of the space Lp(M), p ∈ [1,∞], on which it is considered.

2.6.2. Sturm’s theorem for graphs. A graph (b, c) over (X,m)with an intrinsic metric ρ is said to have uniform subexponential growthif for any ε > 0 there is C > 0 such that for all r > 0 and all x ∈M

m(Br(x)) ≤ Ceεrm(x).

The proof of the following theorem follows the strategy of Sturm in[Stu94].

Theorem 2.13 (Theorem 1 in [BHK13]). Let (b, c) be a connectedgraph over (X,m) and let ρ be an intrinsic metric such that the ballsare finite (B), which has finite jump size (J) and the graph has uniformsubexponential growth. Then,

σ(Lp) = σ(L2), p ∈ [1,∞].

Remark. (a) A question in the direction of p-independence of thespectrum for graphs was already brought up by Davies [Dav07, p. 378].

(b) In contrast to Sturm’s result for manifolds, no curvature typeassumption is needed in the theorem above. Indeed, there are graphswith unbounded weighted vertex degree which satisfy the assumptions,see [BHK13, Example 3.2]. On the other hand, the assumptions ofthe theorem already imply that the combinatorial vertex degree mustbe bounded, see [BHK13, Lemma 3.1].

(c) The statement of the theorem is, in general, wrong if one dropsthe growth assumption. This was already discussed in Section 1.3.2.On the other hand, it is an open question what happens for graphs thatare subexponentially growing, i.e., µ = 0, but not uniformly subexpo-nentially growing.

CHAPTER 3

Curvature on planar tessellations

In this chapter, we survey results relating curvature bounds, geom-etry and spectral theory that are proven in the original manuscripts[Kel10, Kel11, KP11, BGK13, BHK13]. Our focus lies on infiniteplanar tessellations which can be considered as discrete analogues ofnon compact surfaces. The tiles of the tessellations shall be seen asregular polygons.

We study a curvature function that arises as an angular defectand satisfies a Gauß Bonnet formula. This idea goes back at leastto Descartes, see [Fed82], and appeared since then independently atvarious places, see e.g. [Sto76, Gro87, Ish90, Woe98]. A system-atic study of geometric properties of tessellations with non-positivecurvature was undertaken by Baues/Peyerimhoff [BP01, BP06]. Fur-thermore, a substantial amount of research was conducted on vari-ous topics for tessellations in dependence of the curvature, see e.g.[Blo10, Che09, CC08, DM07, Hig01, HJ, HJL, Kel10, KP11,

Oh13, Sto76, SY04, Woe98, Zuk97]. The operators of interestare graph Laplacians with standard weights. First, we show spec-tral bounds resulting from curvature bounds. Here, the quantitativebounds result from estimates on an isoperimetric constant and a vol-ume growth rate, see [KP11]. Secondly, we take a closer look at thecase of uniformly unbounded negative curvature. This is equivalent todiscreteness of spectrum, [Kel10], and we present eigenvalue asymp-totics [BGK13] in this case. Thirdly, we summarize results on thep-independence of the spectrum of the Laplacian as an operator on `p,p ∈ [1,∞], from [BHK13]. Parts of the exposition of this chapter aretaken from the survey article [Kel14b].

One can also define a related notion of curvature for general planargraphs. By the virtue of [Kel11] one sees that non-positive curvatureimplies that the graph is almost a tessellation (possibly with unboundedtiles intersecting in a path of edges). With these considerations mostresults for tessellations can be extended to general planar graphs. Asthis approach is more technical and at some points less geometricallyintuitive, we only discuss it at the end.

53

54 3. CURVATURE ON PLANAR TESSELLATIONS

3.1. Set up and definitions

In this section we introduce planar tessellations, notions of curva-ture and recall the definition of the graph Laplacian.

3.1.1. Planar tessellations. In this chapter we consider graphswith standard weights. So, we adapt our notation of the previouschapters to the notation that is classically used in this context.

The vertex set X is still a countable discrete set. Let b be a graphwith standard weights over X. That is, b takes values in 0, 1 and thefunction c vanishes. We introduce the set of edges as subsets of X withtwo elements as follows

E = x, y ⊆ X | b(x, y) = 1.

A graph is called planar if there is an orientable topological surfaceS that is homeomorphic to R2 such that the graph can be embeddedwithout self intersections into S. The vertices X are mapped to pointsin S and the edges E to line segments in S connecting vertices.

In the following we will identify a combinatorial planar graph withits embedding and denote it by (X,E). Nevertheless, we stress thatwe only use the combinatorial properties of the graph which do notdepend on the embedding.

A graph is locally compact if there is an embedding into S such thatfor every compact K ⊆ S, one has

#e ∈ E | e ∩K 6= ∅ <∞.

Next, we introduce the set of faces F that has the connected com-ponents of

S \⋃

E

as elements. For f ∈ F , we denote by f the closure of f in S.We write G = (X,E, F ) for locally compact planar graphs and,

following [BP01, BP06], we call G = (X,E, F ) a tessellation if thefollowing three assumptions are satisfied:

(T1) Every edge is contained in two faces.(T2) Two faces are either disjoint or intersect in a vertex or an edge.(T3) Every face is homeomorphic to the unit disc.

There are related definitions such as semi-planar graphs see [HJ, HJL]and locally tessellating graphs [Kel11]. Indeed, most of the resultspresented here hold for general planar graphs on surfaces of finite genus.However, the definition of curvature becomes more involved and someof the estimates turn out to be more technical. We refer to Section 3.5for corresponding considerations for planar graphs.

3.1. SET UP AND DEFINITIONS 55

3.1.2. Curvature. Let G = (X,E, F ) be a tessellation. In orderto define a curvature function, we first introduce the vertex degree andthe face degree. We denote the vertex degree of a vertex v ∈ X by

|v| = deg(v) = #edges emanating from v.

We use the notation |v| if we use vertex degree geometrically and deg(v)if we use it analytically. The face degree of a face f ∈ F is defined as

|f | = #boundary edges of f = #boundary vertices of f.

The vertex curvature κ : X → R is defined as

κ(v) = 1− |v|2

+∑

f∈F,v∈f

1

|f | .

The idea traces back to Descartes [Fed82] and was later introduced inthe above form by Stone in [Sto76] who refers to ideas of Alexandrov.Since then, this notion of curvature reappeared at various places, e.g.[Gro87, Ish90] and was widely used, see e.g. [BP01, BP06, DM07,

Hig01, HJL, Kel10, Kel11, KP11, Oh13, Woe98, Zuk97].The notion of curvature is motivated by an angular defect: Assume

a face f is a regular polygon. Then, the inner angles of f are all equalto

β(f) = 2π|f | − 2

2|f | .

This formula is easily derived: Walking around f once results in anangle of 2π, while going around the |f | corners of f one takes a turnby an angle of π − β(f) each time. In this light, the vertex curvaturemay be rewritten as

2πκ(v) = 2π −∑

f∈F,v∈f

β(f), v ∈ X.

It shall be stressed that the mathematical nature of κ is purely com-binatorial. Nevertheless, thinking of the tessellation with a suitableembedding allows for a geometric interpretation. The notion has itsfurther justification in the Gauß-Bonnet formula relating the sum ofthe curvatures of a simply connected set to the Euler characteristic.This formula is mathematical folklore and may, for instance, be foundin [BP01] or [Kel11].

We next consider a finer notion of curvature. Asking which contri-bution to the total curvature at a vertex v comes from the corner at aface f with v ∈ f gives rise to the corner curvature. Precisely, the setof corners of a tessellation G is given by

C(G) = (v, f) ∈ X × F | v ∈ f.


Define the corner curvature κC : C(G)→ R by

κC(v, f) =1

|v| −1

2+

1

|f | .

One immediately infers

κ(v) =∑

f∈F,v∈f

κC(v, f).

This notion of curvature was first introduced in [BP01] and furtherstudied in [BP06, Kel11].

3.1.3. The Laplacians. Next, we recall the definition of the Lapla-cian in the special case of standard weights. Note that planar tessella-tions are special cases of these graphs.

The general quadratic form for graphs with standard weights isgiven by Q : C(X)→ [0,∞]

Q(f) =1

2

∑

v∼w|f(v)− f(w)|2,

and as above we denote the space of functions f in C(X) such thatQ(f) <∞ by D.

As discussed in Section 1.1.3.5, there are two “canonical” measuresfor graphs with standard weights. There is the counting measure whichmeasures the volume of a set by counting the number the vertices. Onthe other hand, there is the degree measure deg which “counts” edgesin a set W ⊆ X. This can be seen by the identity

deg(W ) = 2#EW + #∂W,

where EW are the edges with both vertices in W and ∂W are the edgeshaving one vertex in W and one in X \W . The identity above tells usthat deg(W ) counts the edges with both end vertices in W twice andthe edges leading out once.

The counting measure gives rise to the Hilbert space `2(X) of com-plex valued functions whose square is summable. The scalar producton `2(X) is given by

〈f, g〉 =∑

v∈Xf(v)g(v), f, g ∈ `2(X),

and the norm by ‖f‖ = 〈f, f〉 12 . By the discussion in Section 1.1.3.1the restriction Q to the subspace

D ∩ `2(X) = f ∈ `2(X) | Q(f) <∞.yields a closed positive quadratic form. By Theorem 1.14 we see thatthis form, denoted by Q(N) in Section 1.1.3.1, coincides with the formQ = Q(D) whose domain is the closure of the finitely supported func-tions Cc(X) with respect to ‖ · ‖Q. Hence, the finitely supported func-tions are dense in the form domain.

3.2. CURVATURE AND THE BOTTOM OF THE SPECTRUM 57

Let ∆ be the positive selfadjoint operator associated to Q. Then,∆ acts as

∆f(v) =∑

w∼v(f(v)− f(w))

and by Theorem 1.14 it has the domain

D(∆) = f ∈ `2(X) | ∆f ∈ `2(X).By Theorem 1.5 the operator ∆ is bounded if and only if

supv∈X|v| <∞.

If one equips X with the degree measure deg, then the quadraticform Q restricted to the Hilbert space `2(X, deg) with scalar product

〈f, g〉deg =∑

v∈Xf(v)g(v) deg(v), f, g ∈ `2(X, deg),

is bounded by Theorem 1.5. The associated operator ∆n, the normal-ized Laplacian, is then a bounded operator `2(X, deg) and acts as

∆nf(v) =1

deg(v)

∑

w∼v(f(v)− f(w)).

Recall that the subscript n stems from normalizing measure n whichequals deg in the case of standard weights.

3.2. Curvature and the bottom of the spectrum

In this section, we apply the general theory of the previous chaptersto get explicit estimates for the bottom of the spectrum. First, weconsider a lower bound that follows from an isoperimetric inequalityand then an upper bound that follows from an estimate of the volumegrowth. The results of this section are proven in [KP11].

3.2.1. Lower bounds. Recall the isoperimetric constant α intro-duced in Section 1.5.1 and used in Section 1.5.5:

α = infW⊆X finite

#∂W

deg(W ).

In the case where the face degree is bounded by some q and thevertex degree is bounded by some p the following constant Cp,q ≥ 1will enter the estimate of the isoperimetric constant below

Cp,q :=

1 : if q =∞,1 + 2

q−2: if q <∞ and p =∞,

(1 + 2q−2

)(1 + 2(p−2)(q−2)−2

) : if p, q <∞.


Theorem 3.1 (Theorem 1 in [KP11]). Let G be a tessellation suchthat |v| ≤ p for all v ∈ X and |f | ≤ q for all f ∈ F with p, q ∈ [3,∞].Assume κ < 0 and let K := infv∈X − 1

|v|κ(v). Then

α ≥ 2Cp,qK.

A key idea for the proof is a formula which is attributed in [BP01]to Harm Derksen which is state in the lemma below. It is an immediateconsequence of the Gauß-Bonnet theorem and direct calculation. Wecall a set W ⊆ V simply connected if W and V \W are connected.

Lemma 3.2. [BP01, Proposition 2.1.] Let W ⊆ V be a finite simplyconnected subset of a planar tessellation. Then,

∑

v∈Wκ(v) = 1− #∂W

2+

∑

f∈F,f∩W 6=∅,f∩(V \W )6=∅

#(f ∩W )

|f | .

To get spectral estimates we need the following notation and ob-servations. Let

d = infv∈X|v| and D = sup

v∈X|v|.

The inequality

dλ0(∆n) ≤ λ0(∆)

follows directly from the Rayleigh-Ritz characterization of the bottomof the spectrum, confer [Kel11].

Using this inequality together with Theorem 1.23 we obtain thefollowing corollary from Theorem 3.1.

Corollary 3.3. Let G be a tessellation such that |v| ≤ p for allv ∈ X and |f | ≤ q for all f ∈ F with p, q ∈ [3,∞]. Assume κ < 0 andlet K := infv∈X − 1

|v|κ(v). Then

λ0(∆n) ≥ (1−√

1− 4C2p,qK

2) ≥ 2K2,

and

λ0(∆) ≥ d(1−√

1− 4C2p,qK

2) ≥ 2dK2,

Remark. (a) The two inequalities on the right hand side in thecorollary follow by the Taylor expansion of the square root and Cp,q ≥ 1.(b) The theorem above can be considered as a discrete analogue to atheorem of McKean [McK70] who proves for an n-dimensional com-plete Riemannian manifold M with upper sectional curvature bound−k that the bottom of the spectrum of the Laplace-Beltrami ∆M ≥ 0satisfies

λ0(∆M) ≥ (n− 1)2k/4.

3.2. CURVATURE AND THE BOTTOM OF THE SPECTRUM 59

(c) A fact noted by Higuchi [Hig01], see also [Zuk97], is that if κ <0, then already κ ≤ −1/1806. This extremal case is assumed for atriangle, a heptagon and a 43-gon meeting in a vertex. This impliesthat if κ < 0, then K > 0 and, therefore, λ0(∆n) > 0 and λ0(∆) > 0.This recovers results of [Dod84, Hig01, Woe98].

3.2.2. Upper bounds. In this section we discuss volume growthbounds for tessellations whose face degree is constantly q. We call suchtessellations q-face regular. As a consequence, this yields upper boundsfor the bottom of the essential spectrum.

Denote by Sr the vertices with combinatorial graph distance r ≥ 0to a center vertex o. We will suppress the dependence on o in notationsince it is not important for connected graphs. Furthermore, let Br =⋃rk=0 Sk. We use the upper exponential volume growth µ = µn defined

in Section 2.5.1.2:

µ = lim supn→∞

1

rlog #Br.

The result will be stated in terms of normalized average curvaturesover spheres

κr := κ(Sr) :=

(2q

q − 2

)1

#Srκ(Sr).

Note that the constant 2π(q − 2)/2q is the internal angle of a regularq-gon.

First, we present a volume growth comparison theorem which is ananalogue to the Bishop-Guenther-Gromov comparison theorem fromthe Riemannian setting.

Theorem 3.4 (Theorem 3 in [KP11]). Let G = (X,E, F ) and G =

(X, E, F ) be two q-face regular tessellations with non-positive vertex

curvature, Sr ⊂ X and Sr ⊂ X be spheres with respect to the centers

o ∈ X and o ∈ X, respectively. Assume that the normalized averagespherical curvatures satisfy

κ(Sr) ≤ κ(Sr) ≤ 0, r ≥ 0.

Then the difference sequence (#Sr −#Sr) satisfies #Sr −#Sr ≥ 0, ismonotone non-decreasing and, in particular, we have

µ(G) ≥ µ(G).

We furthermore get an explicit recursion formula for the growthin terms of the normalized average spherical curvatures. This resultcan be proven for tessellations without cut locus. That is for every vthe distance function d(·, v) has no local maxima. For example, this isimplied by non-positive corner curvature [BP06, Theorem 1]. In ourcase of face regular graphs, non-positive corner curvature is equivalent


to non-positive curvature. However, the theorem below is not restrictedto the non-positive curvature case.

For 3 ≤ q <∞ let N = q−22

if q is even and N = q − 2 if q is odd,and

bl =

4q−2− 2 : if q is odd and l = N−1

2,

4q−2

: else,

for 0 ≤ l ≤ N − 1.

Theorem 3.5 (Theorem 2 in [KP11]). Let G = (X,E, F ) be a q-face regular tessellation without cut locus. Then we have the following(N + 1)-step recursion formulas for r ≥ 1

#Sr+1 =

∑r−1l=0 (bl − κ(Sr−l))#Sr−l + #S1 : if r < N ,

∑N−1l=0 (bl − κ(SN−l))#SN−l : if r = N ,

∑N−1l=0 (bl − κ(Sr−l))#Sr−l −#Sr−N : if r > N .

Idea of the proof. The proof given in [KP11] uses strongly theassumption of constant face degree. We count the number of faces crjthat intersect a ball Br in j vertices, where 1 ≤ j ≤ q and q is theconstant face degree. If j ≤ q − 2, then crj is equal to the number of

faces cr+1j+2 that intersect the ball Br+1 in j+2 vertices. Finally, one has

to relate the numbers crj to the number of vertices in a sphere Sr.

The (N + 1)-step recursion formula in the theorem above givesrise to a recursion matrix Mr, r ≥ 0, mapping RN to RN such thatMr(#Sr−N , . . . ,#Sr) = (#Sr−N+1, . . . ,#Sr+1).

In the special case when also the vertex degree is constant, say p,we have a (p, q)-regular tessellation. Then, the constant bl−κk is equalto p− 2, except for l = (N − 1)/2 and q odd. In particular, there is amatrix M such that M = Mr, r ≥ 0. The characteristic polynomial ofM is then given by the complex polynomial

gp,q(z) = 1− (p− 2)z − · · · − (p− 2)zN + zN+1,

if q is even, and

gp,q(z) = 1− (p− 2)z − · · · − (p− 4)zN+1

2 − · · · − (p− 2)zN + zN+1,

if q is odd. By [CW92] and [BCS02], gp,q is a reciprocal Salem polyno-mial, i.e., its roots lie on the complex unit circle except for two positivereciprocal real zeros

1

xp,q< 1 < xp,q < p− 1.

This yields

µ = log xp,q

3.3. DECREASING CURVATURE AND DISCRETE SPECTRUM 61

in the special case of (p, q)-regular tessellation. In particular, the con-siderations above recover the results of Cannon and Wagreich [CW92]and Floyd and Plotnick [FP87, Section 3] that the growth function isa rational function.

Now, we combine these insights with the discrete version of Brook’stheorem by Fujiwara [Fuj96a]

λess0 (∆n) ≤ 1− 2eµn/2

eµn + 1

and the observation

λess0 (∆) ≤ D∞λ

ess0 (∆n)

with D∞ = supK⊆X finite infv∈X\K |v|, to get the following estimate onthe bottom of the essential spectrum of ∆n and ∆.

Theorem 3.6. Let G be a q-face regular tessellation such that

κ(v) ≤ p(1

p− 1

2+

1

q

)≤ 0, v ∈ X,

for some integer p ≥ 3. Then

λess0 (∆n) ≤ 1− 2x

1/2p,q

xp,q + 1

and

λess0 (∆) ≤ D∞

(1− 2x

1/2p,q

xp,q + 1

),

where xp,q is the largest real zero of gp,q above.

3.3. Decreasing curvature and discrete spectrum

In this section, we study the case of uniformly decreasing curvature.More precisely, we look at tessellations where

κ∞ = infK⊆Xfinite

supv∈X\K

κ(v)

equals −∞. For this case, we discuss that the spectrum of ∆n is dis-crete except for the point 1 and the spectrum of ∆ consists only ofdiscrete eigenvalues which accumulate at ∞. In this case, we denotethe eigenvalues of ∆ in increasing order counted with multiplicity byλj(∆), j ≥ 0.

3.3.1. Discrete spectrum. First, we address the spectrum of ∆n.As a bounded operator, ∆n has non empty essential spectrum. In[Kel10, Theorem 5] it was discussed that if the essential spectrum of∆n consists of one point then this point must be 1.

Theorem 3.7 (Theorem 3 (a) in [Kel10]). Let G be a tessellation.The essential spectrum of ∆n consists only of the point 1 if κ∞ = −∞.


It can be seen by examples that the converse implication does nothold in general.

As the operator ∆ is unbounded, it may have empty essential spec-trum. The next theorem characterize this case.

Theorem 3.8 (Theorem 3 (b) in [Kel10]). Let G be a tessellation.The spectrum of ∆ is purely discrete if and only if κ∞ = −∞.

Idea of the proof. If the spectrum of ∆ is purely discrete, thenby the Perrson type theorem 〈∆ϕn, ϕn〉 → ∞ for every normalizedsequence (ϕn) in `2(V ) converges weakly to zero. For a sequence (vn)of vertices and the delta functions δvn , one finds

〈∆δvn , δvn〉 = |vn| ≤ −2κ(vn).

This implies κ∞ = −∞.Now, assume κ∞ = −∞. By Theorem 3.1 we infer that outside oflarge enough finite sets the isoperimetric constant is uniformly positive.Moreover, by results as Theorem 1.23 the bottom of the spectrum of theoperator restricted to functions supported outside of larger and largerfinite sets converges to∞. This, however, is equivalent to pure discretespectrum, as the restricted operators are finite rank perturbations ofthe original operator.

Remark. (a) The theorems above can be considered as discreteanalogues of a theorem of Donnelly/Li [DL79]. This theorem statesthat, for a negatively curved, complete Riemannian manifold M withsectional curvature bound decaying uniformly to −∞, the Laplace-Beltrami operator ∆M has purely discrete spectrum.

(b) In [Fuj96b] Fujiwara proved the statement of Theorem 3.7 forthe normalized Laplacian ∆n on trees.

(c) Wojciechowski [Woj08] showed discreteness of the spectrumof ∆ on general graphs with standard weights in terms of a differentquantity which is sometimes referred to as a mean curvature (see alsothe discussion in Section 1.4).

3.3.2. Eigenvalue asymptotics. An important observation inthe proof of the theorem above is the following estimate

−|v|2≤ κ(v) ≤ 1− |v|

6, v ∈ X.

That implies that | · | and −κ go simultaneously to ∞.In particular, if κ∞ = −∞, then there is a bijective map N0 → X,

j 7→ vj, such that

|vj| ≤ |vj+1|, j ≥ 0.

In [BGK13] it was observed that planar graphs are sparse. Hence,the results of Section 1.5.4 can be used to obtain the following eigen-value asymptotics.

3.5. CURVATURE ON PLANAR GRAPHS 63

Theorem 3.9. Let G be a tessellation. If κ∞ = −∞, then

limj→∞

λj(∆)

|vj|= 1

3.4. The `p spectrum

Now, we turn to the spectrum of the Laplacians as operators on`p(X, deg) and `p(X), p ∈ [1,∞].

For the normalized Laplacian ∆n, consider the generalized Lapla-cian Ln on C(X) with the same mapping rule. By Theorem 1.5, we find

that the restriction ∆(p)n of Ln to `p(X, deg), p ∈ [1,∞] is a bounded

operator. It can easily be seen that ∆(p)n coincides with the generator

of the extension of the semigroups e−t∆n to `p(X, deg), p ∈ [1,∞) and

∆(∞)n being the adjoint of ∆

(n)n .

Simultaneously, let L be the generalized Laplacian as is an exten-sion of ∆ on C(X). Then, it can be seen by Theorem 1.13 that therestriction ∆(p) of L to

D(∆(p)) = f ∈ `p(X) | ∆f ∈ `p(X)is the generator of the extension of the semigroup e−t∆ to `p(X), p ∈[1,∞), and ∆(∞) is the adjoint of ∆(1).

A famous question brought up by Simon [Sim80] and answeredby Hempel/Voigt [HV86] for Schrodinger operators is whether thespectrum depends on the underlying Banach space. Sturm, [Stu93],addressed this question for uniformly elliptic operators on manifoldsin terms of uniform subexponential volume growth. As a special case,he considers curvature bounds. We already discussed the analogueof the general result of Sturm obtained in [BHK13] in Section 2.6.As a consequence of this theorem and some geometric and functionalanalytic ingredients, one can derive the following theorem which istaken from in [BHK13].

Theorem 3.10. (a) If κ ≥ 0, then σ(∆(2)) = σ(∆(p)) for p ∈ [1,∞].(b) If −K ≤ κ < 0, then λ0(∆(2)) 6= λ0(∆(1)).(c) If κ∞ = −∞, then σ(∆(2)) = σ(∆(p)) for all p ∈ (1,∞).

3.5. Curvature on planar graphs

We close this thesis by some considerations on curvature for generalplanar graphs. This was investigated in [Kel11].

For a general planar graph, we have to extend the definitions ofdegrees of faces and vertices. For a corner (v, f) ∈ C(G), we denote by|(v, f)| the minimal number of times the vertex v is met by a boundarywalk of f . Then, we define, for v ∈ X and f ∈ F ,

|v| =∑

(v,g)∈C(G)

|(v, g)| and |f | =∑

(w,f)∈C(G)

|(w, f)|.


As the degree of corners in a tessellation is always one, these definitionscoincide with the ones above in the case of tessellations.

We say a face f is unbounded if |f | =∞. A graph is called simpleif |f | ≥ 3 for all f ∈ F . (By the way we defined graphs in this thesis,this is always satisfied since b(x, x) = 0 and b(x, y) ≤ 1.)

With this notation, we define the corner curvature κC : C(G)→ Rby

κC(v, f) =1

|v| −1

2+

1

|f |and the vertex curvature by κ : X → R by

κ(v) =∑

(v,f)∈C(G)

|(v, f)|κC(v, f).

These definitions are consistent with the definition of κC and κ on tes-sellations and they also satisfy a Gauß-Bonnet formula [Kel11, Propo-sition 1].

Next, we look at a generalization of tessellations. We call a face apolygon if it is homeomorphic to the open unit disc and we call it aninfinigon if it is homeomorphic to the upper half space in R2. A planargraph with a locally compact embedding is called locally tessellating ifit satisfies the following conditions:

(T1) Every edge is contained in two faces.(T2*) Two faces are either disjoint or intersect in a vertex or in a

path of edges. If this path consists of more than one edge,then both faces are unbounded.

(T3*) Every face is a polygon or an infinigon.

Here, (T1) is the same as in the tessellation case. This class of graphsincludes tessellations and trees as well as hybrids. In [Kel11] we findthe following theorem which shows that non-positive curvature on pla-nar graphs implies that the graph is almost a tessellation, i.e., it islocally tessellating.

Theorem 3.11 (Theorem 1 in [Kel11]). Let G be a connected,locally finite, planar graph. If κC ≤ 0 or if G is simple with κ ≤ 0 thenG is locally tessellating and infinite.

For the proof we assume the contrary. One isolates finite areas ofthe graphs on which the assumptions (T1), (T2*), (T3*) fail. Such anarea is then copied finitely many times and pasted along its boundaryto be finally embedded into the 2-dimensional unit sphere. Here, theGauß-Bonnet theorem is used to show that there must be some positivecurvature. By taking enough copies this positive curvature can notcome from the vertices where we pasted the graph but from the inside.

Furthermore, in [Kel11, Theorem 2] it is shown that locally tes-sellating graphs can be embedded into tessellations in a suitable way.

3.5. CURVATURE ON PLANAR GRAPHS 65

This way one can carry over results from tessellations to locally tessel-lating graphs and by the theorem above to planar graphs in the caseof non-positive curvature.

Among the geometric applications in the paper are the followingsome of which are extensions of [BP01, BP06]:

• Absence of cut locus, i.e., every distance minimizing path canbe continued to infinity [Kel11, Theorem 3].• A description of the boundary of distance balls [Kel11, The-

orem 4].• Bounds for the growth of distance balls [Kel11, Theorem 5].• Positivity and bounds for an isoperimetric constant constant

[Kel11, Theorem 6].• Empty interior for minimal bigons and Gromov hyperbolicity

[Kel11, Theorem 7].

The first two results are obtained for non-positive curvature and theother three for negative curvature.

Furthermore, there are applications to the spectral theory. Let usmention that the isoperimetric estimates mentioned above yield ana-logues to the results in Section 3.2.1. Simultaneously, the results ofSection 3.3 carry over by the virtue of [Kel11, Theorem 2].

Let us close this section by a result on absence of compactly sup-ported eigenfunctions. For tessellations such a result was proven in[KLPS06]. In [Kel11] a simplified proof is given in the more generalsetting of planar graphs (which are locally tessellating in the case ofnon-positive curvature by what we discussed above).

Theorem 3.12 (Theorem 9 in [Kel11]). Let G be a connected,locally finite, planar graph such that κC ≤ 0. Then, neither ∆ nor ∆n

admit finitely supported eigenfunctions.

While such a result is true in great generality in continuous settings,it can easily be seen that it may even fail when only κ ≤ 0 (or evenκ < 0) is assumed, [BP06, KPP].

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[Woj08] Radoslaw Krzysztof Wojciechowski. Stochastic completeness ofgraphs. ProQuest LLC, Ann Arbor, MI, 2008. Thesis (Ph.D.)–CityUniversity of New York.

[Woj09] Rados law K. Wojciechowski. Heat kernel and essential spectrum ofinfinite graphs. Indiana Univ. Math. J., 58(3):1419–1441, 2009.

[Woj11] Radoslaw Krzysztof Wojciechowski. Stochastically incomplete man-ifolds and graphs. In Random walks, boundaries and spectra, vol-ume 64 of Progr. Probab., pages 163–179. Birkhauser/Springer BaselAG, Basel, 2011.

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Part 2

Original Manuscripts

CHAPTER 4

M. Keller, D. Lenz, Dirichlet forms and stochasticcompleteness of graphs and subgraphs, Journalfur die reine und angewandte Mathematik 2012

(2012), 189–223.

75

J. reine angew. Math., Ahead of Print

DOI 10.1515/CRELLE.2011.122

Journal fur die reine undangewandte Mathematik( Walter de Gruyter

Berlin New York 2011

Dirichlet forms and stochastic completeness ofgraphs and subgraphs

By Matthias Keller and Daniel Lenz at Jena

Abstract. We study Laplacians on graphs and networks via regular Dirichlet forms.We give a su‰cient geometric condition for essential selfadjointness and explicitly deter-mine the generators of the associated semigroups on all lp, 1e p < y, in this case. Wecharacterize stochastic completeness thereby generalizing all earlier corresponding resultsfor graph Laplacians. Finally, we study how stochastic completeness of a subgraph isrelated to stochastic completeness of the whole graph.

Introduction

There is a long history to the study of the heat equation and spectral theory on graphsand networks (see e.g. the monographs [4], [5] and references therein). The correspondingoperators are known as Laplacians, acoustic operators or generators of symmetric Markovprocesses on the graph or network. A substantial part of this literature is devoted to graphsgiving Laplacians, which are bounded on l2. Recently, certain basic questions concerningunbounded Laplacians have received attention. This is the starting point for our paper.More precisely, we use the framework of regular Dirichlet forms in order to

define the Laplacians on networks via forms (Section 1),

study essential selfadjointness (Theorem 6),

determine the generators of the associated semigroups on lp, 1e p < y, undersuitable conditions (Theorem 5),

characterize stochastic completeness (Theorem 1),

investigate the relationship between stochastic completeness of graphs and sub-graphs (Theorem 2, Theorem 3, Theorem 4).

The use of Dirichlet forms allows us to deal with these questions in a rather general setting.In particular, our results seem to extend all earlier corresponding results. Furthermore, we

hope that our results and the thorough discussion of background and context may be usefulin the study of further questions as well.

Let us discuss these topics in more detail: There are recent investigations of essentialselfadjointness of corresponding Laplacians by Jorgensen [17], of stochastic completenessby Dodziuk and Matthai [11], and of both essential selfadjointness and stochastic com-pleteness by Dodziuk [9], Wojciechowski [28] (see [29] as well) and Weber [26]. Theseinvestigations deal with locally finite graphs and the associated operators. While [11], [9]treat bounded Laplacians, [17], [28], [26] do neither assume a uniform bound on the vertexdegree nor a modification of the measure and, accordingly, the resulting Laplacians are notnecessarily bounded. It turns out that all the Laplacians in question are special instances ofgenerators of regular Dirichlet forms on discrete sets. In fact, there is a one-to-one corre-spondence between the regular Dirichlet forms on a discrete set and graphs over this setwith weights satisfying a certain summability condition. This naturally raises the questionto which extent similar results to the ones in [9], [11], [17], [26], [28] also hold for arbitraryregular Dirichlet forms on discrete sets.

Our first result, Theorem 1, characterizes stochastic completeness for all regularDirichlet forms on discrete sets. This generalizes a main result of [28] (see [9], [11], [26] aswell for related results and a su‰cient condition for stochastic completeness), which in turnis inspired by Grigor’yan’s corresponding result for manifolds [15]. Of course, in terms ofmethods our considerations concerning stochastic completeness heavily draw on existingliterature as e.g. Sturm’s [23] for strongly local Dirichlet forms and Grigor’yans results[15] on Riemannian manifolds. A crucial di¤erence, however, is that our Dirichlet formsare not local. In this sense our results can be understood as providing some non-localcounterpart to [23], [15].

It should be emphasized that—unlike the cited literature—we do allow for non van-ishing killing terms. In order to make sense out of a notion of stochastic completeness inthe presence of a killing term we actually have to extend the usual definition. This is doneby our concept of stochastic completeness at infinity ðSCyÞ and stochastic incompletenessat infinity ðSIyÞ. Let us be a bit more precise: Stochastic completeness concerns loss or con-servation of heat. Now, loss of heat may occur for two reasons. One reason is killing withinthe graph by non-vanishing killing term. The other reason is heat transport to ‘infinity’ orthe ‘boundary’ in finite time. This transport to infinity may happen irrespective of presenceof a killing term. It is this transport to infinity which is captured by our notion of stochasticcompleteness at infinity. Of course, in the case of vanishing killing term stochastic com-pleteness and stochastic completeness at infinity agree. Our Theorem 1 gives a unified treat-ment of the situation. Note that strengthening of the killing may make the graph actuallymore complete at infinity as discussed in Theorem 2.

Let us also mention strongly related work of Feller [12], [13] and of Reuter [22] deal-ing with uniqueness of Markov process on discrete sets with given weights. While theseworks use di¤erent methods and seem to have been somewhat neglected in the above men-tioned literature, they in fact cover parts of the abstract results on stochastic completenessdiscussed in [28], [26]. They are in some sense even more general in that they do not assumesymmetry of the Markov process. We will discuss this more specifically after the statementof our corresponding result. However, we stress already here that a crucial part of ourresult is not covered by [12], [22] as we allow for both a killing term and for arbitrarymeasures on our underlying set.

2 Keller and Lenz, Dirichlet forms and stochastic completeness of graphs and subgraphs

Let us emphasize that our treatment requires intrinsically more e¤ort than theconsiderations of [9], [11], [26], [28] as in our setting the Laplacians (i.e., generators of theDirichlet forms) are known much less explicitly. In fact, in general not even the functionswith compact support will be in the domain of definition of our Laplacians.

As the functions with compact support need not belong to the domain of definition ofour Laplacians, the question of essential selfadjointness does in general not make sensein our context. On the other hand, if the functions with compact support belong to thedomain of definition and a certain geometric condition—called ðAÞ below—is satisfied,we can prove essential selfadjointness of the Laplacians in question on the set of functionswith compact support (Theorem 6). This result extends the corresponding result of [17], [9],[26], [28] to all regular Dirichlet forms on discrete sets. Note that this (again) includes thepresence of an arbitrary killing term and an arbitrary measure on our discrete set. We alsogive examples in which essential selfadjointness fails (as does condition ðAÞ).

Along our way, we can also determine the generators for the corresponding semi-groups on all lp, p A ½1;yÞ, for all regular Dirichlet forms on graphs satisfying ðAÞ. Thesegenerators turn out to be the ‘‘maximal’’ ones (Theorem 5). These results seem to be neweven in the situations considered in [12], [9], [11], [17], [22], [26], [28].

After these considerations, our final aim is to study how ðSCyÞ of a subgraph isrelated to ðSCyÞ of the whole graph. There, we obtain two results: We show that any graphis a subgraph of a graph satisfying ðSCyÞ. This completion can be achieved both by addingkilling terms (Theorem 2) and by adding edges (Theorem 3). We also show that stochasticincompleteness of a suitably modified subgraph implies stochastic incompleteness of thewhole graph (Theorem 4). These results seem to be new even in the contexts discussedearlier.

We have tried to make this paper as accessible and self-contained as possible for bothpeople with a background in Dirichlet forms and people with a background in geometry.For this reason some arguments are given, which are certainly well known.

For further studies of certain spectral features of Laplacians in the framework devel-oped below we refer the reader to [16], [19], both of which were written after the presentpaper.

The paper is organized as follows. In Section 1 we present the notation and our mainresults. A study of basic properties of Dirichlet forms on graphs can be found in Section 2.In Section 3 we consider Dirichlet forms on graphs satisfying the condition ðAÞ mentionedabove. For these forms we calculate the generators of the lp semigroups for p A ½1;yÞ andwe show essential selfadjointness of the generators on l2 (whenever the functions with com-pact support are in the domain of definition). In Section 4 we give examples where essentialselfadjointness fails as well as examples of non-regular Dirichlet forms on graphs. A shortdiscussion of the heat equation in our framework is given in Section 5. Section 6 deals withextending the semigroup and resolvent in question to a somewhat larger space of functions.In Section 7 we can then prove our result characterizing stochastic completeness for arbi-trary Dirichlet forms on graphs. Section 8 contains a proof that any graph is a subgraph ofa stochastically complete graph and that any graph can be made stochastically complete byadding a killing term. Section 9 contains an incompleteness criterion.

3Keller and Lenz, Dirichlet forms and stochastic completeness of graphs and subgraphs

1. Framework and results

Throughout V will be a countable set. Let m be a measure on V with full support(i.e. m is a map m : V ! ð0;yÞ). Then, ðV ;mÞ is a measure space. We will deal exclusivelywith real valued functions. Thus, lpðV ;mÞ, 1e p < y, is defined by

nu : V ! R :

Px AV

mðxÞjuðxÞjp < yo:

Obviously, l2ðV ;mÞ is a Hilbert space with inner product given by

hu; vi :¼P

x AV

mðxÞuðxÞvðxÞ and norm kuk :¼ hu; ui12:

Moreover we denote by lyðVÞ the space of bounded functions on V . Note that this spacedoes not depend on the choice of m. It is equipped with the supremum norm k ky.

A symmetric non-negative form on ðV ;mÞ is given by a dense subspace D of l2ðV ;mÞcalled the domain of the form and a map

Q : D D ! R

with Qðu; vÞ ¼ Qðv; uÞ and Qðu; uÞf 0 for all u; v A D. Such a map is already determined byits values on the diagonal. For u A l2ðV ;mÞ we then define QðuÞ by QðuÞ :¼ Qðu; uÞ if u A D

and QðuÞ :¼ y otherwise. If l2ðV ;mÞ ! ½0;y, u 7! QðuÞ, is lower semicontinuous, Q iscalled closed. If Q has a closed extension, it is called closable and the smallest closed exten-sion is called the closure of Q.

A map C : R ! R with Cð0Þ ¼ 0 and jCðxÞ CðyÞje jx yj is called a normal con-traction. If Q is both closed and satisfies QðCuÞeQðuÞ for all u A l2ðV ;mÞ and all normalcontractions C, it is called a Dirichlet form on ðV ;mÞ (see [3], [6], [14], [20] for backgroundon Dirichlet forms).

Let CcðVÞ be the space of finitely supported functions on V . A Dirichlet Q

form on ðV ;mÞ is called regular if DðQÞXCcðVÞ is both dense in CcðVÞ with respectto the supremum norm and dense in DðQÞ with respect to the form norm given by

k kQ :¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik k2 þ QðÞ

q. As discussed below, for such a regular form the set CcðVÞ is

necessarily contained in the form domain. Thus, a Dirichlet form Q on ðV ;mÞ is regular ifand only if it is the closure of its restriction to the subspace CcðVÞ.

Regular Dirichlet forms on ðV ;mÞ are given by graphs on V , as we discuss next (seeSection 2 for details). A symmetric weighted graph over V or a symmetric Markov chainon V is a pair ðb; cÞ consisting of a map b : V V ! ½0;yÞ with bðx; xÞ ¼ 0 for all x A V

and a map c : V ! ½0;yÞ satisfying the following two properties:

(b1) bðx; yÞ ¼ bðy; xÞ for all x; y A V .

(b2)P

y AV

bðx; yÞ < y for all x A V .


We can then think of ðb; cÞ or rather the triple ðV ; b; cÞ as a weighted graph with vertexset V in the following way: An x A V with cðxÞ3 0 is then thought to be connected to thepoint y by an edge with weight cðxÞ. Moreover, x; y A V with bðx; yÞ > 0 are thought to beconnected by an edge with weight bðx; yÞ. The map b is called the edge weight. The map c

is called killing term. Vertices x; y A V with bðx; yÞ > 0 are called neighbors. More gener-ally, x; y A V are called connected if there exist x0; x1; . . . ; xn; xnþ1 A V with bðxi; xiþ1Þ > 0,i ¼ 0; . . . ; n, and x0 ¼ x, xnþ1 ¼ y. This allows us to define connected components of V inthe obvious way.

To ðV ; b; cÞ we associate the form Qcomp ¼ Qcompb; c on CcðVÞ with diagonal

Qcomp : CcðVÞ ! ½0;yÞ given by

QcompðuÞ ¼ 1

2

Px;y AV

bðx; yÞuðxÞ uðyÞ

2 þP

x AV

cðxÞuðxÞ2:

Obviously, Qcomp is a restriction of the form Qmax ¼ Qmaxb; c;m defined on l2ðV ;mÞ with

diagonal given by

QmaxðuÞ ¼ 1

2

Px;y AV


2 þP

x AV

cðxÞuðxÞ2:

Here, the value y is allowed. It is not hard to see that Qmax is closed and hence Qcomp isclosable on l2ðV ;mÞ (see Section 2) and the closure will be denoted by Q ¼ Qb; c;m and itsdomain by DðQÞ which is the closure of CcðVÞ with respect to k kQ. Then, there exists aunique selfadjoint operator L ¼ Lb; c;m on l2ðV ;mÞ such that

DðQÞ ¼ Domain of definition of L1=2

and

QðuÞ ¼ hL1=2u;L1=2ui

for u A DðQÞ (see e.g. [6], Theorem 1.2.1). As Q is non-negative so is L. Moreover, it is nothard to see that QmaxðCuÞeQmaxðuÞ for all u A l2ðV ;mÞ (and in fact any function u) andevery normal contraction C. Theorem 3:1:1 of [14] then implies that Q also satisfiesQðCuÞeQðuÞ for all u A l2ðV ;mÞ and hence is a Dirichlet form. By construction it is reg-ular. In fact, every regular Dirichlet form on ðV ;mÞ is of the form Q ¼ Qb; c;m (see Theorem7 in Section 2).

Remark. Our setting generalizes the setting of [9], [11], [17], [26], [28] to Dirichletforms on countable sets. In our notation, the situation of [11], [26], [28] can be describedby the assumptions m1 1, c1 0, and bðx; yÞ A f0; 1g for all x; y A V with x3 y and thesetting of [9], [17] can be described by m1 1, c1 0 and bðx; yÞ ¼ 0 for all but finitelymany y for each x A V . In particular, unlike [9], [11], [17], [26], [28] we do not assume finite-ness of the sets fy A V : bðx; yÞ > 0g for all x A V .

Let now a measure m on V with full support and a weighted graph ðb; cÞ over V begiven. Let Q be the associated form and L its generator. Then, by standard theory [7], [14],[20], the operators of the associated semigroup etL, tf 0, and the associated resolvent

aðL þ aÞ1, a > 0, are positivity preserving and even markovian.


Positivity preserving means that they map non-negative functions to non-negativefunctions. In fact, if ðV ; b; cÞ is connected they are even positivity improving, i.e., mapnon-negative nontrivial functions to positive functions (see Section 2). Markovian meansthat they map non-negative functions bounded by one to non-negative functions boundedby one.

This can be used to show that semigroup and resolvent extend to all lpðV ;mÞ,1e pey. These extensions are consistent, i.e., two of them agree on their commondomain and they are selfadjoint, i.e., the adjoint to the extension to lpðV ;mÞ with1e p < y is given by the extension to lqðV ;mÞ for 1=p þ 1=q ¼ 1, see [6]. The corre-sponding generators are denoted by Lp. Thus, the extension of ðL þ aÞ1 to lpðV ;mÞ isgiven by ðLp þ aÞ1. In particular we have L ¼ L2.

We can describe the action of the operator Lp explicitly (in Section 2) as follows(see Theorem 9): Define the formal Laplacian ~LL ¼ ~LLb; c;m on the vector space

~FF :¼n

u : V ! R :P

y

jbðx; yÞuðyÞj < y for all x A Vo

ð1Þ

by

~LLuðxÞ :¼ 1

mðxÞPy


þ cðxÞ

mðxÞ uðxÞ;

where, for each x A V , the sum exists by assumption on u. Then, Lp is a restriction of ~LL forany p A ½1;y.

After having discussed the fact that these are di¤erent semigroups on di¤erent lp

spaces, we will now follow the custom and write etL for all of them.

The preceding considerations show that

0e etL1ðxÞe 1

for all tf 0 and x A V . The question, whether the second inequality is actually an equalityhas received quite some attention. In the case of vanishing killing term, this is discussedunder the name of stochastic completeness or conservativeness. In fact, for c1 0 andbðx; yÞ A f0; 1g for all x; y A V , there is a characterization of stochastic completeness ofWojciechowski [28] (see [9], [11], [26] for related results as well). This characterization isan analogue to corresponding results for Markov processes [12], [22], results on manifoldsof Grigor’yan [15] and results of Sturm for general strongly local Dirichlet forms [23].

Our first main result concerns a version of this result for arbitrary regular Dirichletforms on weighted graphs (see Section 7 for details and proofs concerning the subsequentdiscussion): In this case, we have to replace etL1 by the function

MtðxÞ :¼ etL1ðxÞ þÐt0

esL c

m

ðxÞ ds; x A V :


This is well defined, satisfies 0eM e 1 and for each x A V , the function t 7! MtðxÞ is con-tinuous and even di¤erentiable. Note that for c1 0, we obtain M ¼ etL1 whereas forc3 0 we obtain Mt > etL1 on any connected component of V on which c does not vanishidentically (as the semigroup is positivity improving). The term etL1 can be interpreted asthe amount of heat contained in the graph at time t and the integral can be interpretedas the amount of heat killed within the graph up to the time t. Thus, 1 Mt is the amountof heat transported to the boundary of the graph by the time t and Mt can be interpreted asthe amount of heat, which has not been transported to the boundary of the graph at time t.Our question then becomes whether the quantity

1 Mt

vanishes identically or not. Our result then reads as follows.

Theorem 1 (Characterization of heat transfer to the boundary). Let ðV ; b; cÞ be a

weighted graph and m a measure on V of full support. Then, for any a > 0, the function

w :¼Ðy0

aetað1 MtÞ dt

satisfies 0ewe 1, solves ð~LL þ aÞw ¼ 0, and is the largest non-negative l e 1 with

ð~LL þ aÞl e 0. In particular, the following assertions are equivalent:

(i) For any a > 0 there exists a nontrivial, non-negative bounded l with ð~LL þ aÞl e 0.

(ii) For any a > 0 there exists a nontrivial bounded l with ð~LL þ aÞl ¼ 0.

(iii) For any a > 0 there exists a nontrivial, non-negative bounded l with ð~LL þ aÞl ¼ 0.

(iv) The function w is nontrivial.

(v) MtðxÞ < 1 for some x A V and some t > 0.

(vi) There exists a nontrivial bounded non-negative N : V ½0;yÞ ! ½0;yÞ satisfying

~LLN þ d

dtN ¼ 0 and N0 1 0.

Remark. (a) Conditions (ii) and (iii) deal with eigenvalues of ~LL considered as anoperator on lyðVÞ. In particular, (ii) must fail (for su‰ciently large a) whenever ~LL givesrise to a bounded operator on lyðVÞ. Thus, any bounded operator ~LL yields a stochasticallycomplete graph. In this way we recover the corresponding results of [9], [11].

(b) The case c1 0, m1 1, bðx; yÞ A f0; 1g recovers the corresponding result of [28].In fact, in the case c1 0, m1 1 and general (not even symmetric) b the equivalence of (i)(or (ii)) and (v) is already discussed in [12], [22]. These works mainly aim at studyinguniqueness of the Markov process, i.e., a (somewhat weaker) version of (vi). They charac-terize this uniqueness by validity of (i) for m1 1 and arbitrary c. In this sense it seems fairto say that for c1 0 the equivalence of (i), (v) and (vi) is well known and for general c the


equivalence of (i) and (vi) is well known. Besides our new proof (inspired by [28], [15]), ourmain contribution here is the definition of M allowing for an extension of (v) to situationswith killing terms.

(c) The minimum principle discussed below, Theorem 8, will show that for a > 0 anynontrivial, non-negative solution u of ð~LL þ aÞu ¼ 0 satisfies u > 0 if the underlying graph isconnected.

(d) Let L be the operator associated to a weighted graph ðV ; b; cÞ and L0 the opera-tor associated to ðV ; b; 0Þ, both with respect to the same measure m : V ! ð0;yÞ. Theequivalence of (i) and (v) in the theorem above obviously implies Mt ¼ 1 wheneveretL0 1 ¼ 1, since ~LLl f ~LL0l for every non-negative l A ~FF .

The previous theorem suggests the following definition for stochastic completeness atinfinity and stochastic incompleteness at infinity for general Dirichlet forms on weightedgraphs.

Definition 1.1. The weighted graph ðV ; b; cÞ with the measure m of full support issaid to satisfy ðSIyÞ if it satisfies one (and thus all) of the equivalent assertions of Theorem1. Otherwise ðV ; b; cÞ is said to satisfy ðSCyÞ.

Remark. Note that validity of ðSIyÞ depends on both ðV ; b; cÞ and m. In fact, forgiven ðV ; b; cÞ it is always possible to choose m in such a way that ~LL becomes a boundedoperator on lyðVÞ. Then, ðSCyÞ holds (by (a) of the previous remark).

The following two results show how graphs can be made to satisfy ðSCyÞ by additionof killing terms or edges. They seem to be new even in the setting considered in [9], [11],[26], [28].

Theorem 2. Let m be a measure on V with full support. For any weighted graph

ðV ; b; cÞ there is c 0 : V ! ½0;yÞ such that ðV ; b; c þ c 0Þ satisfies ðSCyÞ.

Remark. Of course, addition of killing terms yields to loss of mass from the graphreflected in the inequality etL1 < 1. As our concept of ðSCyÞ only considers mass trans-ported to the geometric boundary of the graph, we can have and even enforce ðSCyÞ byadding killing terms. More precisely, the theorem can be understood in the following way:Adding a killing term kills heat within the graph on any vertex where the killing term doesnot vanish. If we eliminate enough heat by the killing terms, we can achieve that no moreheat is transferred to the geometric boundary of the graph.

A subgraph ðW ; bW ; cW Þ of a weighted graph ðV ; b; cÞ is given by a subset W of V

and the restriction bW of b to W W and the restriction cW of c to W . The weightedgraph ðV ; b; cÞ is then called a supergraph to ðW ; bW ; cW Þ. Given a measure m on V wedenote its restriction to W by mW . The subgraph ðW ; bW ; cW Þ then gives rise to a formon l2ðW ;mW Þ with associated operator LbW ; cW ;mW

.

Theorem 3. Any weighted graph is the subgraph of a weighted graph satisfying

ðSCyÞ. This supergraph can be chosen to have vanishing killing term if the original graph

has vanishing killing term.


Remark. Note that (in the common definitions) the volume growth of balls in agraph dominates the volume growth of balls in any of its subgraph. Thus, the theoremhas the consequence that failure of ðSCyÞ can not be inferred from lower bounds on thegrowth of volumes of balls.

While subgraphs do not force incompleteness according to Theorem 3, suitablyadjusted subgraphs do force incompleteness of the whole graph. In order to be moreprecise, we need some more notation.

Let ðV ; b; cÞ be a weighted graph with measure m of full support and W a subset of V .Let mW be the restriction of m to W . Let iW : l2ðW ;mW Þ ! l2ðV ;mÞ be the canonicalembedding, i.e., iW ðuÞ is the extension of u to V by setting iW ðuÞ identically zero out-side of W . Let pW : l2ðV ;mÞ ! l2ðW ;mW Þ be the canonical projection, i.e., the adjoint

of iW . Then, W gives rise to the form Qcomp; ðDÞW defined on CcðWÞ by

Qcomp; ðDÞW ðuÞ :¼ QðiW uÞ ¼ Q

compbW ; cW

ðuÞ þP

x AW

dW ðxÞuðxÞ2:

Here, dW ðxÞ :¼P

y AVnW

bðx; yÞ describes the edge deficiency of vertices in W compared to

the same vertex in V . Thus, Qcomp; ðDÞW is in fact the form Qcomp of the weighted graph

ðW ; bðDÞW ; c

ðDÞW Þ with

bðDÞW ¼ bW and c

ðDÞW ¼ cW þ dW :

In particular, by the theory developed above, its closure in l2ðW ;mW Þ, denoted by QðDÞW ,

is a Dirichlet form. The associated selfadjoint operator will be denoted by LðDÞW . This oper-

ator is sometimes thought of as a restriction of the original operator to W with Dirichletboundary condition. For this reason we include the superscript D in the notation. Anotherinterpretation (suggested by the above expression for the form) is to think about the graphwhich arises from the subgraph W by adding one way edges to a vertex at infinity accord-ing to the mentioned edge deficiency.

Again, it is not hard to express the action of LðDÞW explicitly. In fact, the above con-

siderations applied to the graph ðW ; bðDÞW ; c

ðDÞW ;mW Þ show that

LðDÞW u ¼ ~LL

ðDÞW u

for any u A DðLðDÞW Þ. Here, the formal Dirichlet Laplacian ~LL

ðDÞW on W is defined on

pW~FF ¼ i1

w ð ~FFÞ and given by

~LLðDÞW uðxÞ ¼ 1

mðxÞ

Py AW

bðDÞW ðx; yÞ

uðxÞ uðyÞ

þ c

ðDÞW ðxÞuðxÞ

¼ ~LLiW uðxÞ

for x A W . These considerations give that for a function u on W (which is extended by 0to V ) the equality

~LLðDÞW uðxÞ ¼ ~LLuðxÞð2Þ


holds for any x A W . This will be used repeatedly in the sequel. Note also that for W ¼ V

we recover the operator on the whole graph, i.e., ~LLðDÞV ¼ ~LL and L

ðDÞV ¼ L.

The following result seems to be new even in the setting considered in [9], [11], [26],[28].

Theorem 4. Let ðb; cÞ be a weighted graph over V and m a measure on V of full

support. Then ðSIyÞ holds, whenever there exists W LV such that the weighted graph

ðbðDÞW ; c

ðDÞW Þ over the measure space ðW ;mW Þ satisfies ðSIyÞ.

As an example of a situation in which the theorem may be applied we note the fol-lowing consequence.

Corollary 1.2. Let ðb; 0Þ be a weighted graph over V with vanishing killing term and

m a measure on V of full support. Let W be a subset of V such that ðbðDÞW ; 0Þ over the measure

space ðW ;mW Þ satisfies ðSIyÞ and there exists C > 0 withP

y AVnW

bðx; yÞ=mðxÞeC for any

x A W. Then ðb; 0Þ over ðV ;mÞ satisfies ðSIyÞ.

So far, we have not discussed the precise domains of definition for our operators. Infact, the actual domains have been quite irrelevant for our considerations.

To determine the domains we need a geometric condition saying that any infinitepath has infinite measure. More precisely, we define condition ðAÞ as follows:

(A) The equalityP

n ANmðxnÞ ¼ y holds for any sequence ðxnÞ of elements of V such

that bðxn; xnþ1Þ > 0 for all n A N.

Of course, an equivalent requirement would be that the equality mðfxn : n A NgÞ ¼ yholds for any sequence ðxnÞ of pairwise di¤erent elements of V such that bðxn; xnþ1Þ > 0 forall n A N.

Note that ðAÞ is a condition on ðV ;mÞ and ðb; cÞ together. If

infx AV

mðxÞ > 0

holds, then ðAÞ is satisfied for all weighted graphs ðb; cÞ over V .

Our result reads as follows. We are not aware of an earlier result of this form in thiscontext.

Theorem 5. Let ðV ; b; cÞ be a weighted graph and m a measure on V of full support

such that ðAÞ holds. Then, for any p A ½1;yÞ the operator Lp is the restriction of ~LL to

DðLpÞ ¼ fu A lpðV ;mÞ : ~LLu A lpðV ;mÞg:

Remark. The theory of Jacobi matrices already provides examples showing thatwithout ðAÞ the statement becomes false for p ¼ 2. This is discussed in Section 4.


The condition ðAÞ does not imply that ~LLf belongs to l2ðV ;mÞ for all f A CcðVÞ.However, if this is the case, then ðAÞ does imply essential selfadjointness. In this case, Q isthe ‘‘maximal’’ form associated to the weighted graph ðb; cÞ. More precisely, the followingholds.

Theorem 6. Let V be a set, m a measure on V with full support, ðb; cÞ a weighted

graph over V and Q the associated regular Dirichlet form. Assume ~LLCcðVÞL l2ðV ;mÞ.Then, DðLÞ contains CcðVÞ. If furthermore ðAÞ holds, then the restriction of L to CcðVÞ is

essentially selfadjoint and the domain of L is given by

DðLÞ ¼ fu A l2ðV ;mÞ : ~LLu A l2ðV ;mÞg

and the associated form Q satisfies Q ¼ Qmax, i.e.,

QðuÞ ¼ 1

2

Px;y AV


2 þP

x AV

cðxÞuðxÞ2

for all u A l2ðV ;mÞ.

Remark. (a) If inf mðxÞ > 0 then both ðAÞ and ~LLCcðVÞL l2ðV ;mÞ hold for anyweighted graph ðb; cÞ over V . In this case, we recover the corresponding results of [17],[26], [28] on essential selfadjointness, as these works assume m1 1. (They also have addi-tional restrictions on b but this is not relevant here).

(b) The statement on the form being the maximal one seems to be new even in thecontext of [17], [26], [28].

(c) Essential selfadjointness fails in general if ðAÞ does not hold as can be seen byexamples (see Section 4 and the previous remark).

2. Dirichlet forms on graphs—basic facts

In this section we consider a countable set V together with a measure m of fullsupport.

Lemma 2.1. Let Q be a regular Dirichlet form on ðV ;mÞ. Then, CcðVÞ is contained in

DðQÞ.

Proof. Let x A V be arbitrary. Choose j A CcðVÞ with jðxÞ ¼ 2 and jðyÞ ¼ 0 for ally3 x. As CcðVÞXDðQÞ is dense in CcðVÞ with respect to the supremum norm, there existsc A DðQÞ with cðxÞ > 1 and jcðyÞj < 1 for all y3 x. As Q is a Dirichlet form, DðQÞ isinvariant under taking modulus and we can assume that c is non-negative. As Q is aDirichlet form, also ~cc :¼ c51 belongs to DðQÞ. (Here,5denotes the minimum.) As DðQÞis a vector space it contains c ~cc and this is a (nonzero) multiple of j by construction.As x A V was arbitrary, the statement follows. r

Lemma 2.2. Let Q be a regular Dirichlet form on ðV ;mÞ. Then, there exists a

weighted graph ðb; cÞ over V such that the restriction of Q to CcðVÞ equals Qcompb; c .


Proof. By the previous lemma, CcðVÞ is contained in DðQÞ. Then, for any finiteK LV , the restriction QK of Q to CcðKÞ is a Dirichlet form as well. By standard results(see e.g. [1], Theoreme 1), there exists then bK , cK with QK ¼ Q

compbK ; cK

. For K LK 0 and

x; y A K it is not hard to see that bKðx; yÞ ¼ bK 0 ðx; yÞ and cKðxÞf cK 0 ðxÞ. Thus, a simplelimiting procedure gives the result. r

Lemma 2.3. Let m be a measure on V of full support. Let ðb; cÞ be a weighted graph

over V. Then, Qmaxb; c;m is closed and Q

compb; c is closable and its closure Qb; c;m is a restriction

of Qmaxb; c;m.

Proof. It su‰ces to show that Qmaxb; c;m is closed. Thus, it su‰ces to show lower semi-

continuity of u 7! Qmaxb; c;mðu; uÞ. This follows easily from Fatou’s lemma. r

Theorem 7. The regular Dirichlet forms on ðV ;mÞ are exactly given by the forms

Qb; c;m with weighted graphs ðb; cÞ over V.

Proof. By the previous lemma and the discussion in Section 1, any Qb; c;m is a regu-lar Dirichlet form. The converse follows from the previous lemmas. r

The study of regular Dirichlet forms on ðV ;mÞ is based on first understanding theirrestrictions to finite sets. This is done next.

Proposition 2.4. Let ðV ;mÞ be given and ðb; cÞ a weighted graph over V. Let K HV

be finite. Then, LðDÞK is a bounded operator with

LðDÞK f ðxÞ ¼ 1

mðxÞ

Py AK

bðx; yÞ

f ðxÞ f ðyÞþ P

y AVnK

bðx; yÞ þ cðxÞ

f ðxÞ:

In particular, ~LLiK f ðxÞ ¼ LðDÞK f ðxÞ for all x A K, where iK : l2ðK;mKÞ ! l2ðV ;mÞ is the

canonical embedding by extension by zero.

Proof. Every linear operator on the finite dimensional l2ðK;mKÞ is bounded. Thus,we can directly read o¤ the operator L

ðDÞK from the form Q

ðDÞK given by Q

ðDÞK ðuÞ :¼ QðiKuÞ.

This gives the first claim. The last statement follows easily. r

We now discuss two results on solutions of the associated di¤erence equation. Theseresults will be rather useful for our further considerations. We start with a version of aminimum principle.

Theorem 8 (Minimum principle). Let ðV ; b; cÞ be a weighted graph and m a measure

on V of full support. Let U LV be given. Assume that the function u on V satisfies:

ð~LL þ aÞuf 0 on U for some a > 0.

The negative part uU :¼ ujU50 of the restriction of u to U attains its minimum on

each connected component of U.

uf 0 on VnU.

Then, u1 0 or u > 0 on each connected component of U. In particular, uf 0.


Proof. Without loss of generality we can assume U is connected. If u > 0 thereis nothing left to show. It remains to consider the case that there exists x A U withuðxÞe 0. As the negative part of u on U attains its minimum, there exists then xm A U

with uðxmÞe 0 and uðxmÞe uðyÞ for all y A U . As uðyÞf 0 for y A U c, we obtainuðxmÞ uðyÞe 0 for all y A V . By the supersolution assumption we find

0eP

bðxm; yÞuðxmÞ uðyÞ

þ cðxmÞuðxmÞ þ amðxmÞuðxmÞe 0:

As b and c are non-negative and a > 0, we find 0 ¼ uðxmÞ and uðyÞ ¼ uðxmÞ ¼ 0 for all y

with bðy; xmÞ > 0. As U is connected, iteration of this argument shows u1 0 on U . r

The following lemma will be a key tool in our investigations. Note that its proof israther simple due to the discreteness of the underlying space.

Lemma 2.5 (Monotone convergence of solutions). Let a A R, f : V ! R and

u : V ! R be given. Let ðunÞ be a sequence of non-negative functions on V belonging to the

set ~FF given in (1) on which ~LL is defined. Assume un e unþ1 for all n A N, and unðxÞ ! uðxÞand ð~LL þ aÞunðxÞ ! f ðxÞ for all x A V. Then, u belongs to ~FF as well and the equation

ð~LL þ aÞu ¼ f holds.

Proof. Without loss of generality we assume m1 1. By assumption

ð~LL þ aÞunðxÞ ¼P

y AV

bðx; yÞunðxÞ unðyÞ

þcðxÞ þ a

unðxÞ

converges to f ðxÞ for any x A V . AsP

y AV

bðx; yÞunðxÞ converges increasingly to

uðxÞP

y AV

bðx; yÞ < y, the assumptions on un show thatP

y AV

bðx; yÞunðyÞ must converge

as well and in fact must converge toP

y AV

bðx; yÞuðyÞ. From this we easily obtain thestatement. r

We next discuss some fundamental properties of regular Dirichlet forms. These prop-erties do not depend on the graph setting. They are true for general Dirichlet forms andcan, for example, be found in the works [24], [25]. For the convenience of the reader weinclude short proofs based on the previous minimum principle.

Proposition 2.6 (Domain monotonicity). Let ðV ; b; cÞ be a weighted graph and m a

measure of full support. Let K1;K2 LV finite with K1 LK2 be given. Then, for any x A K1,

ðLðDÞK1 þ aÞ1

f ðxÞe ðLðDÞK2

þ aÞ1f ðxÞ

for all f A l2ðV ;mÞ with f f 0 and supp f LK1.

Proof. Consider f A l2ðV ;mÞ with f f 0 and supp f LK1 and define

ui :¼ ðLðDÞKi

þ aÞ1f ; i ¼ 1; 2.

Extending ui by zero we can assume that ui are defined on the whole of V . Then,

ð~LL þ aÞui ¼ f on Ki


for i ¼ 1; 2. Therefore, w :¼ u2 u1 satisfies

w ¼ u2 f 0 on K c1 .

The negative part of w attains its minimum on K1 (as K1 is finite).

ð~LL þ aÞw ¼ f f ¼ 0 on K1.

The minimum principle yields wf 0 on V . r

Regularity is crucial for the proof of the following result.

Proposition 2.7 (Convergence of resolvents/semigroups). Let ðV ; b; cÞ be a weighted

graph, m a measure on V with full support and Q the associated regular Dirichlet

form. Let ðKnÞ be an increasing sequence of finite subsets of V with V ¼S

Kn. Then,

ðLðDÞKn

þ aÞ1f ! ðL þ aÞ1f , n ! y for any f A l2ðK1;mK1Þ. (Here, ðLðDÞ

Knþ aÞ1f is

extended by zero to all of V .) The corresponding statement also holds for the semigroups.

Proof. By general principles (see e.g. [27], Satz 9.20b) it su‰ces to consider theresolvents. After decomposing f in positive and negative part, we can restrict attentionto f f 0. Define un :¼ ðLðDÞ

Knþ aÞ1

f . Then, un f 0. Now, by standard characterization ofresolvents (see e.g. [14], Section 1.4), un is the unique minimizer of

QKnðuÞ þ a u 1

af

2

:

By domain monotonicity, the sequenceunðxÞ

is monotonously increasing for any x A V .

Moreover, by standard results on Dirichlet forms (see e.g. [14], Theorem 1.4.1), we have

un e1

ak f ky and by the spectral theorem kunke

1

ak f k. Thus, the sequence un converges

pointwise and in l2ðV ;mÞ towards a function u A l2ðV ;mÞ. Let now w A CcðVÞ be arbi-trary. Assume without loss of generality that the support of w is contained in K1. Then,QðwÞ ¼ QKn

ðwÞ for all n A N. Closedness of Q, convergence of the ðunÞ and the minimizingproperty of each un then give

QðuÞ þ a u 1

af

2

e lim infn!y

QðunÞ þ a u 1

af

2

¼ lim infn!y

QðunÞ þ a un 1

af

2 !

¼ lim infn!y

QKnðunÞ þ a un

1

af

2 !

e lim infn!y

QKnðwÞ þ a w 1

af

2 !

¼ QðwÞ þ a w 1

af

2

:


As w A CcðVÞ is arbitrary and Q is regular (!), this implies

QðuÞ þ a u 1

af

2

eQðvÞ þ a v 1

af

2

for any v A DðQÞ. Thus, u is a minimizer of

QðuÞ þ a u 1

af

2

:

By characterization of resolvents again, u must then be equal to ðL þ aÞ1f . r

We can use the previous result to connect the operator L to the formal operator ~LL. Todo so we need one further result.

Lemma 2.8. Let ðV ;mÞ be given and ðb; cÞ a weighted graph over V. Let p A ½1;y be

given. For any g A lpðV ;mÞ, the function u :¼ ðLp þ aÞ1g belongs to the set ~FF given in (1)

on which ~LL is defined and solves ð~LL þ aÞu ¼ g.

Proof. We first consider the case p ¼ 2. If su‰ces to consider the case gf 0.Choose an increasing sequence ðKnÞ of finite subsets of V with

SKn ¼ V and let gn be the

restriction of g to Kn. Then, ðgnÞ converges monotonously increasing to g in l2ðV ;mÞ andconsequently ðL þ aÞ1

gn converges monotonously increasing to u. Thus, by monotoneconvergence of solutions (Lemma 2.5), we can assume without loss of generality that g

has compact support contained in K1. By convergence of resolvents, un :¼ ðLðDÞKn

þ aÞ1g

then converges increasingly to u :¼ ðL þ aÞ1g: Moreover, by Proposition 2.4, un satisfies

ð~LL þ aÞun ¼ g on Kn. Thus, the statement follows by monotone convergence of solutions.

We now turn to general p A ½1;y. Again, it su‰ces to consider the case gf 0.Choose an increasing sequence ðKnÞ of finite subsets of V with

SKn ¼ V and let gn be the

restriction of g to Kn. Then, un :¼ ðLp þ aÞ1gn converges to u. Moreover, as gn belongs to

l2ðV ;mÞ consistency of the resolvents gives un ¼ ðL þ aÞ1gn. Now, on the l2ðV ;mÞ levelwe can apply the considerations for p ¼ 2 to obtain

ð~LL þ aÞun ¼ ð~LL þ aÞðL þ aÞ1gn ¼ gn:

Taking monotone limits now yields the statement. r

After these preparations, we can now give the desired information on the generators.

Theorem 9. Let ðV ; b; cÞ be a weighted graph and m a measure on V of full support.

Let p A ½1;y be given. Then, Lp f ¼ ~LLf for any f A DðLpÞ.

Proof. Let f A DðLpÞ be given. Then, g :¼ ðLp þ aÞ f exists and belongs to lpðV ;mÞ.By the previous lemma, f ¼ ðLp þ aÞ1

g solves

ð~LL þ aÞ f ¼ g ¼ ðLp þ aÞ f

and we infer the statement. r


We also note the following by-product of our investigation (see [26], [28], [7] for thisresult for locally finite graphs).

Corollary 2.9 (Positivity improving). Let ðV ; b; cÞ be a connected weighted graph and

L the associated operator. Then, both the semigroup etL, tf 0, and the resolvent ðL þ aÞ1,a > 0, are positivity improving (i.e., they map non-negative nontrivial l2-functions to strictly

positive functions).

Proof. By general principles it su‰ces to consider the resolvent. Let f A l2ðV ;mÞwith f f 0 be given and consider u :¼ ðL þ aÞ1

f . Then uf 0 as the resolvent of aDirichlet form is positivity preserving. If u is not strictly positive, there exists an x withuðxÞ ¼ 0. As u is non-negative, u attains its minimum in x. By Lemma 2.8, u satisfies

ð~LL þ aÞu ¼ f f 0. We can therefore apply the minimum principle (with U ¼ V ) to obtainthat u1 0. This implies f 1 0. r

3. Generators of the semigroups on lp and essential selfadjointness on l2

In this section we will consider a symmetric weighted graph ðV ; b; cÞ and a measure m

on V of full support. We will be concerned with explicitly determining the generators of thesemigroups on lp and studying essential selfadjointness of the generator on l2. Both issueswill be tackled by proving uniqueness of solutions on the corresponding lp spaces. The re-sults of this section are not needed to deal with stochastic completeness.

Recall the geometric assumption introduced in the first section:

(A) The equalityP

n ANmðxnÞ ¼ y holds for any sequence ðxnÞ of elements of V such

that bðxn; xnþ1Þ > 0 for all n A N.

The relevance of ðAÞ comes from the following variant of the minimum principle:

Proposition 3.1. Assume ðAÞ. Let a > 0, p A ½1;yÞ and u A lpðV ;mÞ with

ð~LL þ aÞuf 0 be given. Then, uf 0.

Proof. Assume the contrary. Then, there exists an x0 A V with uðx0Þ < 0. By

0e ð~LL þ aÞuðx0Þ ¼1

mðx0ÞP

y AV

bðx0; yÞuðx0Þ uðyÞ

þ cðx0Þ

mðx0Þuðx0Þ þ auðx0Þ

there must exist an x1 connected to x0 with uðx1Þ < uðx0Þ. Continuing in this way, weobtain a sequence ðxnÞ of connected points with uðxnÞ < uðx0Þ < 0. Combining this withðAÞ, we obtain a contradiction to u A lpðV ;mÞ. r

Let us note the following consequence of the previous minimum principle.

Lemma 3.2 (Uniqueness of solutions on lp). Assume ðAÞ. Let a > 0, p A ½1;yÞ and

u A lpðV ;mÞ with ð~LL þ aÞu ¼ 0 be given. Then, u1 0.


Proof. Both u and u satisfy the assumptions of the previous proposition. Thus,u1 0. r

Remark. The situation for p ¼ y is substantially more complicated as can be seenby (part (ii) of) our first theorem.

This lemma allows us to determine the generators whenever ðAÞ holds.

Proof of Theorem 5. Define

~DDp :¼ fu A lpðV ;mÞ : ~LLu A lpðV ;mÞg:

By Theorem 9, we already know Lp f ¼ ~LLf for any f A DðLpÞ. It remains to show~DDp LDðLpÞ: Let f A ~DDp be given. Then, g :¼ ð~LL þ aÞ f belongs to lpðV ;mÞ. Thus,u :¼ ðLp þ aÞ1

g belongs to DðLpÞ. Now, as shown above, see Lemma 2.8, u solvesð~LL þ aÞu ¼ g. Moreover, f also solves this equation. Thus, by the uniqueness of solutionsgiven in Lemma 3.2, we infer f ¼ u and f belongs to DðLpÞ. This finishes the proof. r

We now turn to a study of essential selfadjointness on CcðVÞ. Clearly, the question ofessential selfadjointness on CcðVÞ only makes sense if ~LLCcðVÞL l2ðV ;mÞ. In this context,we have the following result:

Proposition 3.3. Let ðV ;mÞ be given and ðb; cÞ a weighted graph over V. Then, the

following assertions are equivalent:

(i) ~LLCcðVÞL l2ðV ;mÞ.

(ii) For any x A V , the function V ! ½0;yÞ, y 7! bðx; yÞ=mðyÞ, belongs to l2ðV ;mÞ.

In this case, any u A l2ðV ;mÞ belongs to the set ~FF of (1) on which ~LL is defined and the

three sums

Px AV

uðxÞ~LLvðxÞmðxÞ;P

x AV

~LLuðxÞvðxÞmðxÞ

and

1

2

Px;y AV


vðxÞ vðyÞ

þP

x AV

cðxÞuðxÞvðxÞ

converge absolutely and agree for all u A l2ðV ;mÞ and v A CcðVÞ.

Proof. Without loss of generality we assume c1 0. For any x A V define dx : V ! R

by dxðyÞ ¼ 1 if x ¼ y and dxðyÞ ¼ 0 if x3 y.

Obviously, (i) is equivalent to ~LLdx A l2ðV ;mÞ for all x A V . This latter condition caneasily be seen to be equivalent to (ii). This shows the stated equivalence.


Let u A l2ðV ;mÞ be given. Then, for any x A V , Cauchy–Schwarz inequality and (ii)give

Py AV

jbðx; yÞuðyÞje P

y AV

bðx; yÞ2

mðyÞ

1=2 Py AV

uðyÞ2mðyÞ

1=2

< y:ð*Þ

Thus, u belongs to ~FF . To show the statement on the sums, it su‰ces to consider u A l2ðV ;mÞand v ¼ dz, for z A V arbitrary. In this case, the desired statements can easily be reduced tothe question of absolute convergence of

Px;y AV

bðx; yÞuðxÞdzðxÞ andP

x;y AV

bðx; yÞuðxÞdzðyÞ:

This absolute convergence in turn is shown in (*). r

Proof of Theorem 6. As ~LLCcðVÞL l2ðV ;mÞ, we can define the minimal operatorLmin to be the restriction of ~LL to

DðLminÞ :¼ CcðVÞ

and the maximal operator Lmax to be the restriction of ~LL to

DðLmaxÞ :¼ fu A l2ðV ;mÞ : ~LLu A l2ðV ;mÞg:

The previous proposition gives

hu;Lminvi ¼ Qcompb; c ðu; vÞ

for all u; v A CcðVÞ. This extends to give

hu;Lminvi ¼ Qb; c;mðu; vÞ

for all u A DðQÞ and v A CcðVÞ. Thus, Lmin is a restriction of L in this case.

Moreover, the previous proposition gives also

hu;Lminvi ¼P

x AV

~LLuðxÞvðxÞmðxÞ

for all v A CcðVÞ and u A l2ðV ;mÞ. Thus,

Lmin ¼ Lmax:

Thus, essential selfadjointness of Lmin is equivalent to selfadjointness of Lmax. This in turnis equivalent to L ¼ Lmax (as we have LLLmax by Theorem 5). As ðAÞ and Theorem 5yield DðLÞ ¼ fu A l2ðV ;mÞ : ~LLu A l2ðV ;mÞg, we infer L ¼ Lmax and essential selfadjoint-ness of the restriction of L to CcðVÞ (¼ Lmin) follows.


It remains to show the statement on the form. Let Qmax be the maximal form, i.e.,

QmaxðuÞ ¼ 1

2

Px;y AV


2 þP

x AV

cðxÞuðxÞ2

for all u A l2ðV ;mÞ and LQmax the associated operator. Then, another application of theprevious proposition shows

Px AV

~LLuðxÞvðxÞmðxÞ ¼ 1

2

Px;y AV


vðxÞ vðyÞ

þP

x AV

cðxÞuðxÞvðxÞ

¼ Qmaxðu; vÞ

¼ hLQmax u; vi

for all u A DðLQmaxÞ and v A CcðVÞ. This gives that the self-adjoint operator LQmax associ-ated to Qmax satisfies

LQmaxu ¼ ~LLu ¼ Lu

for all u A DðLQmaxÞ. Thus,

LQmax LL:

As L is selfadjoint, we infer LQmax ¼ L and the statement on the form follows. r

4. Some counterexamples

In this section, we first discuss an example showing that without condition ðAÞTheorem 5 and Theorem 6 fail in general. We then present an example of a non-regularDirichlet form on a weighted graph. Note that the choice of the measure plays a crucialrole here.

Example for failure of Theorem 5 and 6 without assumption (A). Let V 1Z. Let (atfirst) every point of Z have measure 1. Consider the bounded operator

D : l2ðZÞ ! l2ðZÞ; ðDjÞðxÞ ¼ jðx 1Þ þ 2jðxÞ jðx þ 1Þ:

It corresponds to the Dirichlet form Qb;0;1 with bðx; yÞ ¼ 1 whenever jx yj ¼ 1. A directcalculation shows that the function

u : V ! R; uðxÞ :¼ elx

is a positive solution to the equation ð~DDþ aÞu ¼ 0 for a ¼ el þ el 2. Obviously, wehave ab 0 for all real l. Now let w A l1ðZÞ, w > 0 and define the measure

m : V ! ð0;yÞ; mðxÞ :¼ min 1;wðxÞu2ðxÞ


and the killing term

c : V ! ½0;yÞ; cðxÞ :¼ max 0;u2ðxÞwðxÞ 1

amðxÞ:

By construction, u then belongs to l2ðZ;mÞ and

a

mþ c

mþ a1 0:

Let ~LL be defined by

~LLvðxÞ :¼ 1

mðxÞP

y AV

bðx; yÞvðxÞ vðyÞ

þ cðxÞ

mðxÞ vðxÞ;

i.e., in a formal sense ~LL ¼ 1

mð~DDþ cÞ. Then, the restriction Lmax of ~LL to

fv A l2ðZ;mÞ : ~LLv A l2ðZ;mÞg

has the eigenvalue a < 0 since

ðLmax þ aÞuðxÞ ¼ a

mþ c

mþ a

uðxÞ ¼ 0:

Consider now the operator L associated with the Dirichlet form Qb; c;m on l2ðZ;mÞ. Ofcourse, L is a positive operator and therefore can not have a negative eigenvalue. More-over, by the results of the previous section, this operator is a restriction of ~LL. This impliesthat u can not belong to DðLÞ and therefore DðLÞ3DðLmaxÞ. Thus, the domain of defini-tion DðLÞ is not given by Theorem 5. In this case, the restriction of ~LL to CcðVÞ is notessentially self-adjoint (as the proof of Theorem 6 showed that otherwise L ¼ Lmax).

Example of a non-regular Dirichlet form on V . We consider connected graphsðV ; b; cÞ with c1 0 and bðx; yÞ A f0; 1g for all x; y A V . As discussed by Dodziuk–Kendall[10] (see [8], [18] as well) in the context of isoperimetric inequalities, any such graph withpositive Cheeger constant a > 0 has the property that

1

2

Px;y AV

bðx; yÞjðxÞ jðyÞ

2f

a2

2

Px AV

dðxÞjðxÞ2

for all j A CcðVÞ, where dðxÞ ¼P

y AV

bðx; yÞ. Let now such a graph be given. Fix an

arbitrary x0 A V . Choose a measure m with support V and mðVÞ ¼ 1. Thus, the constantfunction 1 belongs to l2ðV ;mÞ. Define the form Q by

QðuÞ :¼ 1

2

Px;y AV


2


for all u A l2ðV ;mÞ for which QðuÞ is finite. Obviously, Q is a Dirichlet form and theconstant function 1 satisfies Qð1Þ ¼ 0. Let now jn be any sequence in CcðVÞ convergingto 1 in l2ðV ;mÞ. Then, jnðx0Þ converges to 1. In particular,

QðjnÞfa2

2dðx0Þjnðx0Þ2 ! a2

2dðx0Þ > 0; n ! y:

Thus, QðjnÞ does not converge to 0 ¼ Qð1Þ. Hence, Q is not regular.

5. The heat equation on lT

In this section we consider a weighted graph ðb; cÞ over the measure space ðV ;mÞ withassociated formal operator ~LL.

A function N : ½0;yÞ V ! R is called a solution of the heat equation if for eachx A V the function t 7! NtðxÞ is continuous on ½0;yÞ and di¤erentiable on ð0;yÞ and foreach t > 0 the function Nt belongs to the domain of ~LL and the equality

d

dtNtðxÞ ¼ ~LLNtðxÞ

holds for all t > 0 and x A V . For a bounded solution N continuity of ½0;yÞ ! R,t 7! ~LLNtðxÞ, can easily be seen (for each fixed x A V ). For such N validity ofd

dtNtðxÞ ¼ ~LLNtðxÞ for t > 0 then extends automatically to t ¼ 0, i.e., t 7! NtðxÞ is

di¤erentiable on ½0;yÞ andd

dtNtðxÞ ¼ ~LLNtðxÞ holds for any tf 0.

The following theorem is essentially a standard result in the theory of semigroups. Inthe situation of special graphs it has been shown in [26], [28]. For completeness reason wegive a proof in our situation as well.

Theorem 10. Let L be a self-adjoint restriction of ~LL, which is the generator of a Di-

richlet form on l2ðV ;mÞ. Let v be a bounded function on V and define N : ½0;yÞ V ! R

by NtðxÞ :¼ etLvðxÞ. Then, the function NðxÞ : ½0;yÞ ! R, t 7! NtðxÞ, is di¤erentiable and

satisfies

d

dtNtðxÞ ¼ ~LLNtðxÞ

for all x A V and tf 0.

Proof. As v is bounded, continuity of t 7! NtðxÞ follows from general principleson weak l1-ly continuity of the semigroup on lyðV ;mÞ, see e.g. [6]. It remains to showdi¤erentiability and the validity of the equation.

As discussed already, it su‰ces to consider t > 0. After decomposing v into positiveand negative part, we can assume without loss of generality that v is non-negative.


Let ðKnÞ be a sequence of finite increasing subsets of V withS

Kn ¼ V . Let vn be thefunction on V which agrees with v on Kn and equals zero elsewhere. Thus, vn belongsto l2ðV ;mÞ and we can consider etLvn for any n A N. For each fixed x A V the functiont 7! etLvnðxÞ converges monotonously to t 7! NtðxÞ (by definition of the semigroupon ly). As t 7! NtðxÞ is continuous, this convergence is even uniform on compactsubintervals of ð0;yÞ. Moreover, standard l2-theory shows that Nn ¼ etLvn satisfiesd

dtNnðxÞ ¼ LNnðxÞ for all t > 0 and x A V . By assumption L is a restriction of ~LL and

we infer

d

dtNnðxÞ ¼ 1

mðxÞP

y AV

bðx; yÞNnðxÞ NnðyÞ

cðxÞ

mðxÞNnðxÞ

for all x A V and t > 0. This equality together with the uniform convergence on compactintervals in ð0;yÞ and the summability of the bðx; yÞ in y gives uniform convergence of

thed

dtNnðxÞ on compact intervals. Hence, t ! NtðxÞ is di¤erentiable on ð0;yÞ and satisfies

the desired equation. r

Lemma 5.1. Let N be a bounded solution ofd

dtN þ ~LLN ¼ 0, N0 1 0. Then, for

arbitrary a > 0, the function v :¼Ðy0

etaNt dt solves ð~LL þ aÞv ¼ 0.

Proof. This follows by a short calculation: By boundedness of N andP

y

bðx; yÞ <y,we can interchange two limits to obtain

~LLvðxÞ ¼ limT!y

ÐT0

eta ~LLNtðxÞ dt:

Using that N solves the heat equation and partial integration we find

~LLvðxÞ ¼ limT!y

ÐT0

eta d

dtNtðxÞ

dt

¼ limT!y

etaNtðxÞjT0

ÐT0

aetaNtðxÞ dt

¼ aÐy0

etaNtðxÞ dt

¼ avðxÞ:

Here, we used boundedness of N and N0 ¼ 0 to get rid of the boundary terms after thepartial integration. r

6. Extended semigroups and resolvents

We are now going to extend the resolvents/semigroups to a larger class of func-tions. To do so, we note that for a function f on V with f f 0 the functions g A CcðVÞ


with 0e ge f form a net with respect to the natural ordering g h whenever ge h.Limits along this net will be denoted by lim

g f. As the resolvents and semigroups are positivity

preserving, for f f 0, a > 0, t > 0, we can define the functions ðL þ aÞ1f : V ! ½0;y

and etLf : V ! ½0;y by

ðL þ aÞ1f ðxÞ :¼ lim

g fðL þ aÞ1

gðxÞ

and

etLf ðxÞ :¼ limg f

etLgðxÞ:

In fact, as we are in a discrete setting, the operators have kernels, i.e., for any tf 0 thereexists a unique function

etL : V V ! ½0;yÞ with etLf ðxÞ ¼P

y AV

etLðx; yÞ f ðyÞ

for any f f 0 (and similarly for the resolvent). It is not hard to see that for functions inlyðVÞ, these definitions are consistent with our earlier definitions.

Theorem 11 (Properties of extended resolvents and semigroups). Let a > 0 be given.

Let f be a non-negative function on V.

(a) Let Kn be an increasing sequence of finite subsets of V withS

Kn ¼ V. Let fn be

the restriction of f to Kn, and un :¼ ðLðDÞKn

þ aÞ1fn. Then, un converges pointwise monoto-

nously to ðL þ aÞ1f .

(b) The following statement are equivalent:

(i) There exists a non-negative l : V ! ½0;yÞ with ð~LL þ aÞl f f .

(ii) ðL þ aÞ1f ðxÞ is finite for any x A V.

In this case u :¼ ðL þ aÞ1f is the smallest non-negative function l with ð~LL þ aÞl f f

and it satisfies ð~LL þ aÞu ¼ f .

(c) For all x A V

ðL þ aÞ1f ðxÞ ¼

Ðy0

etaetLf ðxÞ dt:

Remark. Note that the functions in (a) and (c) are allowed to take the value y.Statement (a) is an extension of Proposition 2.7 to non-negative functions.

Proof. Throughout the proof we let u denote the function ðL þ aÞ1f .


(a) Let x A V be given. By domain monotonicity unðxÞ ¼ ðLðDÞKn

þ aÞ1fnðxÞ is increas-

ing. Moreover, again by domain monotonicity and fn e f we have

unðxÞ ¼ ðLðDÞKn

þ aÞ1fnðxÞe ðL þ aÞ1

fnðxÞe ðL þ aÞ1f ðxÞ ¼ uðxÞ

for all n. It remains to show the ‘converse’ inequality. We consider two cases.

Case 1. uðxÞ < y. Let e > 0 be given. By definition of the extended resolvents thereexists then g A CcðVÞ with 0e ge f and

uðxÞ ee ðL þ aÞ1gðxÞ:

As g has compact support, we can assume without loss of generality that the support of g iscontained in Kn for all n. By convergence of resolvents, we conclude

ðL þ aÞ1gðxÞ ee ðLðDÞ

Knþ aÞ1

gðxÞ

for all su‰ciently large n. Thus, for such n we find

uðxÞ 2ee ðLðDÞKn

þ aÞ1gðxÞ:

By ge f and supp gLK1, we have ge fn for all n. Thus, the last inequality gives

uðxÞ 2ee ðLðDÞKn

þ aÞ1fnðxÞ ¼ unðxÞ:

This finishes the considerations for this case.

Case 2. uðxÞ ¼ y. Let k > 0 be arbitrary. By definition of the extended resolventsthere exists then g A CcðVÞ with 0e ge f and

ke ðL þ aÞ1gðxÞ:

Now, we can continue as in Case 1 to obtain

k ee ðLðDÞKn

þ aÞ1fnðxÞ ¼ unðxÞ

for all su‰ciently large n. As k > 0 is arbitrary the statement follows.

(b) We first show (ii) ) (i): Recall that u ¼ ðL þ aÞ1f and consider g A CcðVÞ with

0e ge f . Then, by Lemma 2.8, ug :¼ ðL þ aÞ1g solves

ð~LL þ aÞug ¼ g:

Taking monotone limits on both sides and using the finiteness assumption (ii), we obtain

ð~LL þ aÞu ¼ f :

This shows (i) (with l ¼ u).


We next show (i) ) (ii): Let l f 0 satisfy ð~LL þ aÞl f f . Let ðKnÞ be an increasingsequence of finite subsets of V as in (a) and let fn be the restriction of f to Kn. Extendun :¼ ðLðDÞ

Knþ aÞ1

fn by zero to all of V . Then, wn :¼ l un satisfies:

wn ¼ l f 0 on K cn .

The negative part of wn attains its minimum on Kn (as Kn is finite).

ð~LL þ aÞwn ¼ ð~LL þ aÞl ð~LL þ aÞun f f f ¼ 0 on Kn.

The minimum principle, Theorem 8, then gives

wn ¼ l un f 0:

As n is arbitrary and un converges to u by part (a), we find that ue l is finite. This finishesthe proof of the equivalence statement of (b). The last statements of (b) have already beenshown along the proofs of (i) ) (ii) and (ii) ) (i).

(c) For g A CcðVÞ with 0e ge f the equation

ðL þ aÞ1g ¼

Ðy0

etaetLg dt

holds by standard theory on semigroups. Now, (c) follows by taking monotone limits onboth sides. r

There is a special function v to which our considerations can be applied:

Proposition 6.1. For any a > 0 we have the estimate

0e ðL þ aÞ1 a1 þ c

m

e 1:

Remark. Let us stress that c=m is not assumed to be bounded.

Proof. As a1 þ c=mf 0, we have 0e ðL þ aÞ1ða1 þ c=mÞ. Moreover, we obvi-ously have

ð~LL þ aÞ1 ¼ a1 þ c

m:

Thus, (b) of the previous theorem shows ðL þ aÞ1ða1 þ c=mÞe 1. r

We will also need the following consequence of the proposition.

Proposition 6.2. Let ðV ; b; cÞ be a weighted graph and define S : V ! ½0;y by

SðxÞ :¼Ðy0

esL c

m

ðxÞ ds:

Then, S satisfies 0eS e 1 and ~LLS ¼ c=m.


Proof. For g A CcðVÞ with 0e ge c=m and a > 0, we define Sg;a by

Sg;a :¼Ðy0

easesLgðxÞ ds

and Sg by Sg ¼ lima!0

Sg;a. Then,

Sg;a ¼ ðL þ aÞ1g; i:e:; ð~LL þ aÞSg;a ¼ g:

By ge a1 þ c=m for any a > 0 and Proposition 6.1, we have

Sg;a ¼ ðL þ aÞ1gðxÞe ðL þ aÞ1 a1 þ c

m

ðxÞe 1:

As Sg is the monotone limit of the Sg;a, this shows that Sg is bounded by 1. Moreover,using the uniform bound on the Sg;a and taking the limit a ! 0 in

ð~LL þ aÞSg;a ¼ g;

we find

~LLSg ¼ gf 0:

As S ¼ limgc=m

Sg and the Sg are uniformly bounded, we obtain the statement. r

Lemma 6.3. Let uf 0 be given. Then, the following assertions are equivalent:

(i) etLue u for all t > 0.

(ii) ðL þ aÞ1ue

1

au for all a > 0.

Any uf 0 with ~LLuf 0 satisfies these equivalent conditions.

Proof. The implication (i) ) (ii) follows easily from Theorem 11 (c) giving

ðL þ aÞ1 ¼Ðy0

etaetL dt. Similarly, the implication (ii) ) (i) follows by a limiting

argument from the standard

etLf ¼ limn!y

t

nL þ n

t

!n

f

for f A l2ðV ;mÞ. As for the last statement, we note that ~LLuf 0 implies

1

að~LL þ aÞuf u:

By (b) of Theorem 11 the desired statement (ii) follows. r


7. Characterization of stochastic completeness

In this section, we can finally characterize stochastic completeness. We begin by intro-ducing the crucial quantity in our studies.

Lemma 7.1. Let ðV ; b; cÞ be a weighted graph and m a measure on V with full

support. Then, the function M : ½0;yÞ V ! ½0;y defined by

MtðxÞ :¼ etL1ðxÞ þÐt0

esL c

m

ðxÞ ds

satisfies 0eMs eMt e 1 for all sf tf 0 and, for each x A V , the map t 7! MtðxÞ is

di¤erentiable and satisfiesd

dtMtðxÞ þ ~LLMtðxÞ ¼ cðxÞ=mðxÞ.

Remark. We can give an interpretation of M in terms of a di¤usion process on V

as follows. For x A V , let dx be the characteristic function of fxg. A di¤usion on V starting

in x with normalized measure is then given by1

mðxÞ dx at time t ¼ 0. It will yield to theamount of heat

etL dx

mðxÞ ; 1

¼ dx

mðxÞ ; etL1

¼P

y AV

etLðx; yÞ ¼ etL1ðxÞ

within V at the time t. Moreover, at each time s the rate of heat killed at y by the killingterm c is given by esLðx; yÞcðyÞ=mðyÞ. The total amount of heat killed at y until the time t

is then given byÐt0

esLðx; yÞcðyÞ=mðyÞ ds. The total amount of heat killed at all vertices

by c till the time t is accordingly given by

Py AV

Ðt0

esLðx; yÞ cðyÞmðyÞ ds ¼

Ðt0

Py AV

esLðx; yÞ cðyÞmðyÞ ds ¼

Ðt0

esL c

m

ðxÞ ds:

This means that M measures the amount of heat at time t which has not been transferred tothe boundary of V .

Proof. By definition we have M f 0. By Proposition 6.2, SðxÞ ¼Ðy0

ðesLc=mÞðxÞ ds

is finite and we can therefore calculate

Ðt0

esL c

m

ðxÞ ds ¼ SðxÞ

Ðyt

esL c

m

ðxÞ ds ¼ SðxÞ etLSðxÞ;

where the last statement follows by taking monotone limits along the net of g A CcðVÞ with0e ge c=m. Thus,

Mt ¼ etL1 þ S etLS ¼ S þ etLð1 SÞ:


From this equality the desired statements follow easily: By Proposition 6.2, we have1 S f 0 and ~LLð1 SÞ ¼ ~LL1 ~LLS ¼ c=m c=m ¼ 0. Lemma 6.3 then yields

esLð1 SÞe etLð1 SÞe 1 S

for all sf tf 0. Plugging this into the formula for Mt gives, for all 0e te s,

0eMs eMt e 1:

Moreover, as S and the constant function 1 are bounded, we can apply Theorem 10 toMt ¼ etLð1 SÞ þ S to infer that t 7! MðxÞ is di¤erentiable with

d

dtMtðxÞ ¼ ~LLetL1ðxÞ þ ~LLetLSðxÞ ¼ ~LLMtðxÞ þ ~LLSðxÞ ¼ ~LLMtðxÞ þ

cðxÞmðxÞ ;

where we used ~LLS ¼ c=m from Proposition 6.2. r

We now show that integration over M yields a resolvent.

Lemma 7.2. ðL þ aÞ1ða1 þ c=mÞðxÞ ¼Ðy0

aetaMtðxÞ ds:

Proof. As shown in (c) of Theorem 11 we have

ðL þ aÞ1 a1 þ c

m

ðxÞ ¼

Ðy0

aetaetL 1 þ c

am

ðxÞ dt:

Thus, it su‰ces to show that

Ðy0

eta etL c

m

ðxÞ dt ¼

Ðy0

aeta

Ðt0

esL c

m

ðxÞ ds

dt:

This follows by partial integration applied to each (non-negative) term of the sum

etL c

m

ðxÞ ¼

Py AV

etLðx; yÞ cðyÞmðyÞ :

This finishes the proof. r

Remark. Let us stress that the care taken with monotone convergence in the abovearguments is quite necessary. For example one might think that 1 ¼ ðL þ aÞ1ð~LL þ aÞ1.Combined with the previous lemma, this would lead to

1 ¼ ðL þ aÞ1ða1 þ c=mÞ ¼Ðy0

aetaMt dt:

However, the phenomenon we study is exactly that the integral can be strictly smallerthan 1!


After these preparations we now prove our first main result. Recall that we defined

w ¼Ðy0

aetað1 MtÞ dt:

Proof of Theorem 1. AsÐy0

aeta dt ¼ 1, the previous lemma gives

w ¼ 1 ðL þ aÞ1ða1 þ c=mÞ:

Thus, w solves ð~LL þ aÞw ¼ 0. Moreover, the minimality properties of the extended resol-vent (see Theorem 11(b)) yield the maximality property of w. More precisely, let l be anynon-negative function bounded by 1 with ð~LL þ aÞl e 0. Then, 1 l is non-negative andsatisfies

ð~LL þ aÞð1 lÞ ¼ a1 þ c

m ð~LL þ aÞl f a1 þ c

m:

The minimality property of 1 w ¼ ðL þ aÞ1ða1 þ c=mÞ then gives 1 we 1 l, and thedesired inequality l ew follows.

It remains to show the equivalence statements.

(v) ) (iv): This is clear as 0eMt e 1 and M is continuous.

(iv) ) (iii): This is clear from the properties of w shown above.

(iii) ) (ii): This is clear.

(ii) ) (i): Let lþ be the positive part of l, i.e., lþðxÞ ¼ lðxÞ if lðxÞ > 0 and lþðxÞ ¼ 0otherwise. If lþ is trivial, the function l is a nontrivial, non-negative bounded solutionand (i) follows. Otherwise a direct calculation shows that lþ is a nontrivial subsolution.Obviously, lþ is non-negative and bounded.

(i) ) (v): If there exists a nontrivial non-negative subsolution, then w as the largestsubsolution must be nontrivial. Hence, there must exist t > 0 and x A V with MtðxÞ < 1.

(v) ) (vi): Lemma 7.1 gives that N :¼ 1 M satisfies N0 ¼ 0 andd

dtN þ ~LLN ¼ 0.

This gives the desired implication.

(vi) ) (i): This is a direct consequence of Lemma 5.1. r

8. Stochastically complete graphs with incomplete subgraphs

In this section we prove Theorem 2 and Theorem 3. For Theorem 3 the basic ideais to attach graphs satisfying ðSCyÞ to each vertex of a graph (with ðSIyÞ) such that theresulting graph will satisfy ðSCyÞ. As adding a potential to a graph can be interpreted asadding edges to infinity, the proof of Theorem 2 can be seen as a variant of the proof ofTheorem 3.


The graphs we attach will be given as follows. Let ðN; bN ; 0Þ the graph with vertex setN ¼ f0; 1; 2; . . .g, bNðx; yÞ ¼ 1 if jx yj ¼ 1 and bNðx; yÞ ¼ 0 otherwise and c1 0. More-over, let the measure m on N be constant m1 1. The next lemma shows that when u solvesð~LLN þ aÞuðxÞ ¼ 0 for some a > 0 and all x A Nnf0g then it is only bounded if it is exponen-tially decreasing.

Lemma 8.1. Let ðN; bN ; 0Þ be as above and m1 1. Let a > 0 be given. Let u be a

positive function on N with ð~LLN þ aÞuðxÞ ¼ 0 for all xf 1. If for some xf 1,

uðxÞf 2

2 þ auðx 1Þ;

then u increases exponentially.

Proof. Let u be a positive solution. If 1 þ a

2

uðxÞf uðx 1Þ for some xf 1, we

get by the equation ð~LLN þ aÞuðxÞ ¼ 0:

0 ¼ 1 þ a

2

uðxÞ uðx þ 1Þ þ 1 þ a

2

uðxÞ uðx 1Þ

f 1 þ a

2

uðxÞ uðx þ 1Þ:

This implies uðx þ 1Þf 1 þ a

2

uðxÞ and, in particular, 1 þ a

2

uðx þ 1Þf uðxÞ. By

induction we then get for yf x:

uðyÞf 1 þ a

2

yx

uðxÞ

which gives the statement. r

Proof of Theorem 3. Let ðW ; bW ; cW Þ be a weighted graph and m a measure of fullsupport on W . If ðSCyÞ holds we are done, so we assume the contrary. We will construct aweighted graph ðV ; b; cÞ satisfying ðSCyÞ such that W LV and bjWW ¼ bW . Define

degbWðxÞ ¼ 1

mðxÞP

y AW

bðx; yÞ:

Let n : W!ð0;yÞ be a function which satisfies nðxÞ degbWðxÞmðxÞ A N and

Pyj¼1

nðxjÞ ¼ y

for any sequence ðxjÞ in W . (For example we can set nðxÞ ¼ ½degbWm þ 1=degbW

m, where½x denotes the smallest integer not exceeding x.)

To each vertex x A W , we attach nðxÞ degbWðxÞmðxÞ copies of the weighted graph

ðN; bN ; 0Þ defined in the beginning of the section. We do this by identifying x A W with


the vertices 0 in the associated copies of N. We denote the resulting graph by V and defineb on V V by letting

bðx; yÞ ¼bW ðx; yÞ : x; y A W ;

bNðx; yÞ : x; y in the same copy of N;

0 : otherwise:

8<:

Moreover, we extend c and m to V by letting c1 0 and m1 1 on VnW and denote~LL ¼ ~LLV . We will show that for all a > 0 every non-negative nontrivial function u on V ,which satisfies ð~LL þ aÞu ¼ 0, is unbounded. Without loss of generality, we can assumethat the graph is connected. Then, any non-negative nontrivial solution u of ð~LL þ aÞu ¼ 0must be positive by the minimum principle. Let u be such a positive solution ofð~LL þ aÞu ¼ 0 and assume it is bounded.

Fix x0 A W and a sequence ðrrÞ in R with ð2 þ aÞ=2 > rr > 1 andP

ðrr 1Þ < y.By induction we can now define for each r A N an xr A V such that bðxr; xr1Þ > 0and rruðxrþ1Þf sup

y AV ;bðxr;yÞ>0

uðyÞ. Since we assumed u bounded, Lemma 8.1 gives

uðyÞ < 2uðxrÞ=ð2 þ aÞ for each vertex y in a copy of N which is adjacent to xr. If xrþ1

was in a copy of N, then this would imply that u has a maximum in xr which leads to acontradiction to ð~LL þ aÞu ¼ 0. Thus all xr belong to W . The equation ð~LL þ aÞuðxrÞ ¼ 0now gives

0 ¼ 1

mðxrÞP

y AV

bðxr; yÞuðxrÞ uðyÞ

þ cðxrÞ

mðxrÞuðxrÞ þ auðxrÞ

f degbWðxrÞuðxrÞ þ

1

mðxrÞ

Py AVnW

bðxr; yÞuðxrÞ uðyÞ

P

y AW

bðxr; yÞuðyÞ

f 1 þ anðxrÞ2 þ a

degbW

ðxrÞuðxrÞ rr degbWðxrÞuðxrþ1Þ:

In the second inequality, we used a; cðxrÞ; uðxrÞf 0. In the third inequality, we estimatedthe sum over y A VnW by the inequality uðyÞ < 2=ð2 þ aÞuðxrÞ of Lemma 8.1 and thesum over y A W by the choice of xrþ1. We get by direct calculation and iteration

uðxrþ1Þf1

rr

anðxrÞ2 þ a

þ 1

uðxrÞf

Qrj¼1

1

rj

Qrj¼1

anðxjÞ2 þ a

þ 1

!uðx0Þ:

Letting r tend to infinity the right-hand side diverges if and only if n is chosen such thatPyj¼1

nðxjÞ is divergent. (Notice that the infinite product over ð1=rjÞ is greater than zero since

we assumed that ðrj 1Þ is summable.) Thus, by our choice of n, we arrive at the con-tradiction that u is unbounded. By Theorem 1, this construction shows that for everyðW ; bW ; cW Þ there is a weighted graph ðV ; b; cÞ which is ðSCyÞ and ðW ; bW ; cW Þ is a sub-graph of ðV ; b; cÞ. r

Remark. An alternative construction is to add single vertices instead of copies of N.For the resulting graph and a function u satisfying ð~LL þ aÞu ¼ 0 the value of u on anadded vertex y adjacent to the vertex x in the original graph is then determined by


ð1 þ aÞuðyÞ ¼ uðxÞ. The rest of the proof can now be carried out in a similar manner. Wechose to do the construction above to avoid the impression that the ðSCyÞ is the result ofadding some type of boundary to the graph.

We finish this section by proving Theorem 2.

Proof of Theorem 2. Set bðxÞ :¼P

y AV

bðx; yÞ. Choose c 0 : V ! ½0;yÞ such that for

any sequence ðxjÞ in V satisfying bðxj; xjþ1Þ > 0 for all j A N, we have

Pyj¼1

cðxjÞ þ c 0ðxjÞbðxjÞ

¼ y:

(For example one may choose c 0ðxÞ ¼ bðxÞ for x A V .) We now follow a similar reason-ing as in the proof of Theorem 3: We consider a nontrivial non-negative solution u

of ð~LLb; cþc 0;m þ aÞu ¼ 0, a > 0 and choose inductively for each r A N an xr A V and2f rr > 1 with uðx0Þ > 0, bðxr; xrþ1Þ > 0 and rruðxrþ1Þf sup

y:bðxr;yÞ>0

uðyÞ for all r A N.

Then, a direct calculation gives uðxrþ1Þf1

rr

1 þ cðxrÞ þ c 0ðxrÞbðxÞ

uðxrÞ and unboundedness

of u follows (whenever rr converges to 1 su‰ciently fast). Hence, by Theorem 1 the graphðV ; b; c þ c 0Þ satisfies ðSCyÞ. r

9. An incompleteness criterion

In this section we prove Theorem 4, which is a counterpart to Theorem 3. As shownthere a subgraph with ðSIyÞ is well compatible with the whole graph satisfying ðSCyÞ.Theorem 4 shows under which additional condition ðSIyÞ of a subgraph implies ðSIyÞ forthe whole graph. This condition is about how heavily the incomplete subgraph is connectedwith the rest of the graph. Not having control over the amount of connections leadspossibly to ðSCyÞ as we have seen in Theorem 3.

For a subset W of a weighted graph ðV ; b; cÞ we define the outer boundary qW of W

in V by

qW ¼ fx A VnW : by A W ; bðx; yÞ > 0g:

Note that the outer boundary of W is a subset of VnW . We will be concerned withdecompositions of the whole set V into two sets W and W 0 :¼ VnW . In this case, thereare two outer boundaries. Our intention is to extend positive bounded functions u on W

with ð~LLðDÞW þ aÞue 0 to positive bounded functions v on the whole space satisfying

ð~LL þ aÞve 0. To do so, we will have to take particular care at what happens on the twoboundaries.

Lemma 9.1. Let ðV ; b; cÞ be a connected weighted graph. Let W LV be non-empty.

Then, any connected component of W 0 ¼ VnW contains a point x A qW.

Proof. Choose x A W arbitrarily. By assumption, any y A W 0 is connected to x by apath in V , i.e., there exist x0; x1; . . . ; xn A V with bðxi; xiþ1Þ > 0 and x0 ¼ x, xn ¼ y. Let


m A f0; . . . ; ng be the largest number with xm A W . Then, xmþ1 belongs to both the bound-ary of W and the connected component of y. r

Lemma 9.2. Let ðV ; b; cÞ be a weighted graph and m a measure of full support. Let

U LV and j be a non-negative function in l2ðU ;mÞ. Then, ðLðDÞU þ aÞ1j is non-negative on

U and positive on the connected components of any x A U with jðxÞ > 0.

Proof. The operator LðDÞU is associated to the weighted graph ðU ; b

ðDÞU ; c

ðDÞU Þ. Hence,

Corollary 2.9 gives the statement. r

Lemma 9.3. Let ðV ; b; cÞ be a weighted graph and m a measure of full support. Let

U LV be given. Let v A ~FF and denote the restriction of v to U by u and the restriction of v

to VnU by u 0. Then, for any x A U ,

ð~LL þ aÞvðxÞ ¼ ð~LLðDÞU þ aÞuðxÞ 1

mðxÞP

y AVnU

bðx; yÞu 0ðyÞ:

Proof. This follows by direct calculation. r

Proof of Theorem 4. Let W LV be given such that for every a > 0 there is abounded non-negative nontrivial function u on W satisfying

ð~LLðDÞW þ aÞue 0:

By Theorem 1, it su‰ces to show that any such u can be extended to a non-negative andbounded function v on V such that

ð~LL þ aÞve 0:

To do so, we proceed as follows: Set W 0 ¼ VnW . Define

c : W 0 ! R; cðxÞ ¼ 1

mðxÞP

y AW

bðx; yÞuðyÞ:

Thus, c vanishes on W 0nqW and is non-negative on qW . Now, choose j A l2ðW 0;mW Þwith 0e jec and jðxÞ3 0 whenever cðxÞ3 0. Thus,

jf 0 on qW and j1 0 on W 0nqW :

Define u 0 on W 0 by

u 0 :¼ ðLðDÞW 0 þ aÞ1j:

As j is non-negative on qW , combining Lemma 9.1 and Lemma 9.2 shows that u 0 is non-negative (on W 0). Now, define v on V by setting v equal to u on W and setting v equal to u 0

on W 0. We now investigate for each x A V the value of

ð~LL þ aÞvðxÞ:

We consider four cases.


Case 1: x A WnqW 0. Then, ð~LL þ aÞvðxÞ ¼ ð~LLðDÞW þ aÞuðxÞe 0 by assumption on u.

Case 2: x A W 0nqW . Then, ð~LL þ aÞvðxÞ ¼ ð~LLðDÞW 0 þ aÞu 0ðxÞ ¼ jðxÞ ¼ 0 by construc-

tion of u 0.

Case 3: x A qW 0. Lemma 9.3 with U ¼ W gives

ð~LL þ aÞvðxÞ ¼ ð~LLðDÞW þ aÞuðxÞ

Py AW 0

bðx; yÞu 0ðyÞe 0:

Here, the last inequality follows as ð~LLðDÞW þ aÞuðxÞe 0 by assumption on u and u 0 is non-

negative.

Case 4: x A qW . Lemma 9.3 with U ¼ W 0 gives

ð~LL þ aÞvðxÞ ¼ ð~LLðDÞW 0 þ aÞu 0ðxÞ

Py AW

bðx; yÞuðyÞ ¼ jðxÞ cðxÞe 0:

This finishes the proof. r

Proof of Corollary 1.2. Let a > 0 be given and a constant C f 0 such thatPy AVnW

bðx; yÞ=mðxÞeC for all x A W . A non-negative subsolution for aþ C with respect

to the operator associated to ðbðDÞW ; 0Þ is obviously a subsolution for a with respect to the

operator associated to ðbðDÞW ; c

ðDÞW Þ. By assumption such non-negative, nontrivial, bounded

subsolutions for aþ C and ðbðDÞW ; 0Þ exist for all a > 0. Therefore ðbðDÞ

W ; cðDÞW Þ satisfies ðSIyÞ.

The statement now follows from Theorem 4. r

Acknowledgments. The research of M.K. is financially supported by a grant fromKlaus Murmann Fellowship Programme (sdw). Part of this work was done while he wasvisiting Princeton University. He would like to thank the Department of Mathematics forits hospitality. He would also like to thank Jozef Dodziuk and Radek Wojciechowskifor several inspiring discussions bringing up some of the questions which motivated thispaper. D.L. would like to thank Andreas Weber for most stimulating discussions and PeterStollmann for generously sharing his knowledge on Dirichlet forms on many occasions.The careful reading of the referee and partial support from German Science Foundation(DFG) are gratefully acknowledged.

References

[1] A. Beurling and J. Deny, Espaces de Dirichlet, I, Le cas elementaire, Acta Math. 99 (1958), 203–224.

[2] A. Beurling and J. Deny, Dirichlet spaces, Proc. Natl. Acad. Sci. USA 45 (1959), 208–215.

[3] N. Bouleau and F. Hirsch, Dirichlet forms and analysis on Wiener space, de Gruyter Stud. Math. 14,

de Gruyter, 1991.

[4] F. R. K. Chung, Spectral Graph Theory, CBMS Reg. Conf. Ser. Math. 92, Amer. Math. Soc., 1997.

[5] Y. Colin de Verdiere, Spectres de graphes, Soc. Math. de France, Paris 1998.

[6] E. B. Davies, Heat kernels and spectral theory, Cambridge University Press, Cambridge 1989.

[7] E. B. Davies, Linear operators and their spectra, Cambridge Stud. Adv. Math. 106, Cambridge University

Press, Cambridge 2007.


[8] J. Dodziuk, Di¤erence equations, isoperimetric inequality and transience of certain random walks, Trans.

Amer. Math. Soc. 284 (1984), 787–794.

[9] J. Dodziuk, Elliptic operators on infinite graphs, Analysis, geometry and topology of elliptic operators,

World Sci. Publ., Hackensack, NJ (2006), 353–368.

[10] J. Dodziuk and W. S. Kendall, Combinatorial Laplacians and isoperimetric inequality, in: From local times

to global geometry, control and physics, Pitman Res. Notes Math. 150 (1986), 68–74.

[11] J. Dodziuk and V. Matthai, Kato’s inequality and asymptotic spectral properties for discrete magnetic

Laplacians, in: The ubiquitous heat kernel, Contemp. Math. 398, Amer. Math. Soc. (2006), 69–81.

[12] W. Feller, On boundaries and lateral conditions for the Kolmogorov di¤erential equations, Ann. Math. (2)

65 (1957), 527–570.

[13] W. Feller, Notes to my paper ‘‘On boundaries and lateral conditions for the Kolmogorov di¤erential equa-

tions’’, Ann. Math. (2) 68 (1958), 735–736.

[14] M. Fukushima, Y. Oshima and M. Takeda, Dirichlet forms and symmetric Markov processes, de Gruyter

Stud. Math. 19, de Gruyter, 1994.

[15] A. Grigor’yan, Analytic and geometric background of reccurrence and non-explosion of the brownian motion

on riemannian manifolds, Bull. Amer. Math. Soc. 36 (1999), 135–249.

[16] S. Haeseler and M. Keller, Generalized solutions and spectrum for Dirichlet forms on graphs, in: D. Lenz,

F. Sobieczky, W. Woess, eds., Boundaries and spectra of random walks, Progr. Probab., to appear.

[17] P. E. T. Jorgensen, Essential selfadjointness of the graph-Laplacian, J. Math. Phys. 49 (2008).

[18] M. Keller, Essential spectrum of the Laplacian on rapidly branching tessellations, Math. Ann. 346 (2010),

no. 1, 51–66.

[19] M. Keller and D. Lenz, Unbounded Laplacians on graphs: Basic spectral properties and the heat equation,

Math. Model. Nat. Phenom. 5 (2010), no. 2.

[20] Z. M. Ma and M. Rockner, Introduction to the theory of (non-symmetric) Dirichlet forms, Springer, 1992.

[21] B. Metzger and P. Stollmann, Heat kernel estimates on weighted graphs, Bull. Lond. Math. Soc. 32 (2000),

477–483.

[22] G. E. H. Reuter, Denumerable Markov processes and the associated contraction semigroups on l, Acta

Math. 97 (1957), 1–46.

[23] P. Stollmann, A convergence theorem for Dirichlet forms with applications to boundary value problems with

varying domains, Math. Z. 219 (1995), 275–287.

[24] P. Stollmann and J. Voigt, Perturbation of Dirichlet forms by measures, Potential Anal. 5 (1996), 109–138.

[25] K.-T. Sturm, Analysis on local Dirichlet spaces, I, Recurrence, conservativeness and L p-Liouville properties,

J. reine angew. Math. 456 (1994), 173–196.

[26] A. Weber, Analysis of the physical Laplacian and the heat flow on a locally finite graph, J. Math. Anal. Appl.

370 (2010), 146–158.

[27] J. Weidmann, Lineare Operatoren in Hilbertraumen I, Teubner, Stuttgart 2000

[28] R. K. Wojciechowski, Stochastic completeness of graphs, PHD thesis, 2007, arXiv:0712.1570v2.

[29] R. K. Wojciechowski, Heat kernel and essential spectrum of infinite graphs, Indiana Univ. Math. J. 58 (2009),

1419–1441.

Mathematisches Institut, Friedrich Schiller Universitat Jena, 07743 Jena, Germany

e-mail: [email protected]

Mathematisches Institut, Friedrich Schiller Universitat Jena, 07743 Jena, Germany

e-mail: [email protected]

Eingegangen 18. Marz 2010, in revidierter Fassung 12. Dezember 2010


CHAPTER 5

M. Keller, D. Lenz, Unbounded Laplacians onGraphs: Basic Spectral Properties and the Heat

Equation, Mathematical modeling of naturalphenomena: Spectral Problems 5 (2010), 198–224.

111

Math. Model. Nat. Phenom.Vol. 5, No. 2, 2009, pp.

Unbounded Laplacians on graphs:Basic spectral properties and the heat equation

Matthias Keller1

Daniel Lenz2 ∗

1 Mathematisches Institut, Friedrich Schiller Universitat Jena,D- 07743 Jena, Germany, [email protected].

2 Mathematisches Institut, Friedrich Schiller Universitat Jena,D- 07743 Jena, Germany, [email protected],

URL: http://www.analysis-lenz.uni-jena.de/

Abstract. We discuss Laplacians on graphs in a framework of regular Dirichlet forms. We focuson phenomena related to unboundedness of the Laplacians. This includes (failure of) essentialselfadjointness, absence of essential spectrum and stochastic incompleteness.

Key words: Dirichlet forms, graphs, essential self adjointness, essential spectrum, stochastic com-pletenessAMS subject classification: 47B39, 60J27

IntroductionThe study of Laplacians on graphs is a well established topic of research (see e.g. the monographs[4, 6] and references therein). Such operators can be seen as discrete analogues to Schrodingeroperators. Accordingly their spectral theory has received quite some attention. Such operatorsalso arise as generators of symmetric Markov processes and they appear in the study of heat equa-tions on discrete structures. Recently, certain themes related to unboundedness properties of suchoperators have become a focus of attention. These themes include

• definition of the operators and essential selfadjointness,

• absence of essential spectrum,

• stochastic incompleteness.

∗Corresponding author. E-mail: [email protected]

1

Matthias Keller and Daniel Lenz Unbounded Laplacians on graphs

In this paper we want to survey recent developments and provide some new results. Our principlegoal is to make these topics accessible to non-specialists by providing a somewhat gentle andintroductory discussion.

Let us be more precise. We consider a graph with weights on edges and vertices. The weightscan be seen to give a generalized vertex degree.

There is an obvious way to formally associate a symmetric nonnegative operator to such agraph. If the generalized vertex degrees are uniformly bounded this operator is bounded and allformal expressions make sense. If the generalized vertex degrees are not uniformly bounded al-ready the definition of a self adjoint operator is an issue. This issue can be tackled by provingessential selfadjointness of the formal operator on the set of functions with compact support. Thiswas done for locally finite weighted graphs by Jorgensen in [19] and for locally finite graphs byWojciechowski in [32] and Weber in [31]. These results require local finiteness and do not al-low for weights on the corresponding `2 space. As discussed by Keller/Lenz in [21] it is possibleto get rid of the local finiteness requirement and to allow for weighted spaces by using Dirichletforms. The corresponding results give a nonnegative selfadjoint (but not necessarily essentiallyselfadjoint) operator in quite some generality and provide a criterion for essential selfadjointnesscovering the earlier results of [19, 31, 32]. These topics are discussed in Section 1.

Having an unbounded nonnegative operator at ones disposal one may then wonder about its ba-sic spectral features. These basic features include the position of the infimum of the spectrum andthe existence of essential spectrum. Both issues can be approached via isoperimetric inequalities.In fact, lower bounds for the spectrum have been considered by Dodziuk [9] and Dodziuk/Kendall[11]. For planar graphs explicit estimates for the isoperimetric constant and hence for the spectrumcan be found for instance in [17, 18, 22, 23, 30]. Triviality of the essential spectrum for generalgraphs has been considered by Fujiwara [14]. The corresponding results deal with bounded opera-tors only. (They allow for unbounded vertex degree but then force boundedness of the operators byintroducing weights on the corresponding `2 space.) Still, the methods can be used to provide lowerbounds on the spectrum and prove emptiness of the essential spectrum for unbounded Laplaciansas well. For locally finite graphs this has been done by Keller in [20]. Here, we present a general-ization of the results of [20] to the general setting of regular Dirichlet forms. This generalizationalso extends the results of [14, 11] to our setting. This is discussed in Section 5.

Finally, we turn to a (possible) consequence of unboundedness in the study of the heat equa-tion viz stochastic incompleteness. Stochastic incompleteness describes the phenomenon that massvanishes in a diffusion process. While this may a priori not seem to be connected to unbounded-ness, it turns out to be connected. This has already been observed by Dodziuk/Matthai [12] andDodziuk [10] in that they show stochastic completeness for certain bounded operators on graphs.A somewhat more structural connection is provided by our discussion below. For locally finitegraphs stochastic completeness has recently been investigated by Weber in [31] and Wojciechowski[32]. In fact, Weber presents sufficient conditions and Wojciechowski gives a characterization ofstochastic incompleteness. This characterization is inspired by corresponding work of Grigor’yanon manifolds [16] (see work of Sturm [27] for related results as well). As shown in [21] this char-acterization can be extended to regular Dirichlet forms. Details are discussed in Section 8. There,we also provide some further background extending [21]. Let us mention that this circle of ideas

2


is strongly connected to questions concerning uniqueness of Markov process with given generatoras discussed by Feller in [13] and Reuter in [26]. We take this opportunity to mention the veryrecent survey [34] of Wojciechowski giving a thorough discussion of stochastic incompletenessfor manifolds and graphs (with edge weight constant to one).

While our basic aim is to study unbounded Laplacians we complement our results by charac-terizing boundedness of the Laplacians in question in Section 3.

The paper is organized as follows. In Section 1 we introduce our operators and discuss basicproperties. Section 2 contains a useful minimum principle and some of its consequences. Bound-edness of the Laplacians in question is characterized in Section 3. A useful tool, the so calledco-area formulae are investigated in Section 4. They are used in Section 5 to provide an isoperi-metric inequality which is then used to study bounds on the infimum of the (essential) spectrum.This allows us in particular to characterize emptiness of the essential spectrum. The connectionto Markov processes is discussed in Section 7. A characterization of stochastic incompleteness isgiven in Section 8.

1. Graph Laplacians and Dirichlet formsThroughout V will be a countably infinite set.

1.1. Weighted graphsWe will deal with weighted graphs with vertex set V . A symmetric weighted graph over V is apair (b, c) consisting of a map b : V × V −→ [0,∞) with b(x, x) = 0 for all x ∈ V and a mapc : V −→ [0,∞) satisfying the following two properties:

(b1) b(x, y) = b(y, x) for all x, y ∈ V .

(b2)∑

y∈V b(x, y) <∞ for all x ∈ V .

Then b is called the edge weight and c is called killing term.We consider (b, c) or rather the triple (V, b, c) as a weighted graph with vertex set V in the

following way: An x ∈ V with c(x) 6= 0 is thought to be connected to the point ∞ by an edgewith weight c(x). Moreover, x, y ∈ V with b(x, y) > 0 are thought to be connected by an edgewith weight b(x, y). Vertices x, y ∈ V with b(x, y) > 0 are called neighbors. More generally,x, y ∈ V are called connected if there exist x0, x1, . . . , xn ∈ V with b(xi, xi+1) > 0, i = 0, . . . , nand x0 = x, xn = y. This allows us to define connected components of V in the obvious way.

Two examples have attracted particular attention.

Example (Locally finite graphs): Let (V, b, c) be a weighted graph with c ≡ 0 and b(x, y) ∈0, 1 for all x, y ∈ V . We can then think of the (x, y) ∈ V × V with b(x, y) = 1 as connectedby an edge with weight 1. The condition (b2) then implies that any x ∈ V is connected to only

3


finitely many y ∈ V . Such graphs are known as locally finite graphs. This is the class of examplesstudied in [20, 14, 32, 31].

Example (Locally finite weighted graphs): Let (V, b, c) be a weighted graph with c ≡ 0 andb satisfying

]y : b(x, y) 6= 0 <∞for all x ∈ V . Then, (V, b, c) is called a locally finite weighted graph. This is the class of examplesstudied in [10, 19].

1.2. Dirichlet forms on countable setsLet m be a measure on V with full support (i.e., m is a map m : V −→ (0,∞)). Then, (V,m) is ameasure space. A particular example is given by m ≡ 1. We will deal exclusively with real valuedfunctions. Thus, `p(V,m), 0 < p <∞, is defined by

u : V −→ R :∑

x∈Vm(x)|u(x)|p <∞.

Obviously, `2(V,m) is a Hilbert space with inner product 〈·, ·〉 = 〈·, ·〉m given by

〈u, v〉 :=∑

x∈Vm(x)u(x)v(x) and norm ‖u‖ := 〈u, u〉 12 .

Moreover we denote by `∞(V ) the space of bounded functions on V . Note that this space does notdepend on the choice of m. It is equipped with the supremum norm ‖ · ‖∞ defined by

‖u‖∞ := supx∈V|u(x)|.

A symmetric nonnegative form on (V,m) is given by a dense subspace D of `2(V,m) calledthe domain of the form and a map

Q : D ×D −→ Rwith Q(u, v) = Q(v, u) and Q(u, u) ≥ 0 for all u, v ∈ D.

Such a map is already determined by its values on the diagonal (u, u) : u ∈ D ⊆ D × D.This motivates to consider the restriction of Q to the diagonal as an object on its own right. Thus,for u ∈ `2(V,m) we then define Q(u) by

Q(u) :=

Q(u, u) : u ∈ D,∞ : u 6∈ D.

If `2(V,m) −→ [0,∞], u 7→ Q(u) is lower semicontinuous Q is called closed. If Q has a closedextension it is called closable and the smallest closed extension is called the closure of Q.

A map C : R −→ R with C(0) = 0 and |C(x)−C(y)| ≤ |x−y| is called a normal contraction.If Q is both closed and satisfies Q(Cu) ≤ Q(u) for all normal contractions C and all u ∈ `2(V,m)it is called a Dirichlet form on (V,m) (see [3, 7, 15, 24] for background on Dirichlet forms).

4


Let Cc(V ) be the space of finitely supported functions on V . A Dirichlet form on (V,m) iscalled regular if its domain contains Cc(V ) and the form is the closure of its restriction to thesubspace Cc(V ). (The standard definition of regularity for Dirichlet forms would require thatD(Q) ∩ Cc(V ) is dense in both Cc(V ) and D(Q). As discussed in [21] this is equivalent to ourdefinition.)

1.3. From weighted graphs to Dirichlet formsThere is a one-to-one correspondence between weighed graphs and regular Dirichlet forms. Thisis discussed next.

To the weighted graph (V, b, c) we can then associate the form Qmax = Qmaxb,c,m : `2(V,m) →

[0,∞] with diagonal given by

Qmax(u) =1

2

∑

x,y∈Vb(x, y)(u(x)− u(y))2 +

∑

x∈Vc(x)u(x)2.

Here, the value ∞ is allowed. Let Qcomp = Qcompb,c be the restriction of Qmax to Cc(V ). It is

not hard to see that Qmax is closed. Hence Qcomp is closable on `2(V,m) and the closure will bedenoted by Q = Qb,c,m and its domain by D(Q).

As discussed in [21] (see [15] as well) the following holds.

Theorem 1. The regular Dirichlet forms on (V,m) are exactly given by the forms Qb,c,m withweighted graphs (b, c) over V .

Remark. One may wonder whether the regularity assumption is necessary in the above theorem.It turns out that not every Dirichlet form Qmax

b,c,m is regular. A counterexample is provided in [21].

For a given a weighted graph (V, b, c) the different choices of measure m will produce differentDirichlet forms. Two particular choices have attracted attention. One is the choice of m ≡ 1.Obviously, this choice does not depend on b and c. Another possibility is to use n = m = mb,c

given byn(x) :=

∑

y∈Vb(x, y) + c(x).

The advantage of this measure is that it produces a bounded form (see below for details).

1.4. Graph LaplaciansLet m be a measure on V of full support, (b, c) a weighted graph over V and Qb,c,m the associatedregular Dirichlet form. Then, there exists a unique selfadjoint operator L = Lb,c,m on `2(V,m)such that

D(Q) := u ∈ `2(V,m) : Q(u) <∞ = Domain of definition of L1/2

5


andQ(u) = 〈L1/2u, L1/2u〉

for u ∈ D(Q) (see e.g. Theorem 1.2.1 in [7]). As Q is nonnegative so is L.

Definition 2. Let V be a countable set andm a measure on V with full support. A graph Laplacianon V is an operator L associated to a form Qb,c,m.

Our next aim is to describe the operator L more explicitly: Define the formal Laplacian L =Lb,c,m on the vector space

F := u : V −→ R :∑

y∈V|b(x, y)u(y)| <∞ for all x ∈ V (1.1)

by

Lu(x) :=1

m(x)

∑

y∈Vb(x, y)(u(x)− u(y)) +

c(x)

m(x)u(x),

where, for each x ∈ V , the sum exists by assumption on u. The operator L describes the action ofL in the following sense.

Proposition 3. Let (V, b, c) be a weighted graph and m a measure on V of full support. Then, theoperator L is a restriction of L i.e.,

D(L) ⊆ u ∈ `2(V,m) : Lu ∈ `2(V,m) and Lu = Lu

for all u ∈ D(L).

In order to obtain further information we need a stronger condition. We define condition (A)as follows:

(A) For any sequence (xn) of vertices in V such that b(xn, xn+1) > 0 for all n ∈ N, the equality∑n∈Nm(xn) =∞ holds.

Let us emphasize that in general (A) is a condition on (V,m) and b together. However, if

infx∈V

mx > 0

holds, then obviously (A) is satisfied for all graphs (b, c) over V . This applies in particular to thecase that m ≡ 1.

Given (A) we can say more about the generators [21].

Theorem 4. Let (V, b, c) be a weighted graph and m a measure on V of full support such that (A)

holds. Then, the operator L is the restriction of L to

D(L) = u ∈ `2(V,m) : Lu ∈ `2(V,m).

6


Remark. The theory of Jacobi matrices already provides examples showing that without (A) thestatement becomes false [21].

The condition (A) does not imply that Lf belongs to `2(V,m) for all f ∈ Cc(V ). However,if this is the case, then (A) does imply essential selfadjointness. In this case, Q is the “maximal”form associated to the graph (b, c). More precisely, the following holds [21].

Theorem 5. Let V be a set, m a measure on V with full support, (b, c) a graph over V and Q theassociated regular Dirichlet form. Assume LCc(V ) ⊆ `2(V,m). Then, D(L) contains Cc(V ). Iffurthermore (A) holds, then the restriction of L to Cc(V ) is essentially selfadjoint and the domainof L is given by

D(L) = u ∈ `2(V,m) : Lu ∈ `2(V,m)and the associated form Q satisfies Q = Qmax i.e.,

Q(u) =1

2

∑

x,y∈Vb(x, y)(u(x)− u(y))2 +

∑

x∈Vc(x)u(x)2

for all u ∈ `2(V,m).

Remark. Essential selfadjointness may fail if (A) does not hold as can be seen by examples [21].

If infx∈V mx > 0 then both (A) and LCc(V ) ⊆ `2(V,m) hold for any graph (b, c) over V . Wetherefore obtain the following corollary.

Corollary 6. Let V be a set and m a measure on V with infx∈V mx > 0. Then, D(L) containsCc(V ), the restriction of L to Cc(V ) is essentially selfadjoint and the domain of L is given by

D(L) = u ∈ `2(V,m) : Lu ∈ `2(V,m)

and the associated form Q satisfies Q = Qmax.

Remark. The corollary includes the case that m ≡ 1 and we recover the corresponding results of[11, 32, 31] on essential selfadjointness. (In fact, the cited works also have additional restrictionson b but this is not relevant here.)

2. Minimum principle and consequencesAn important tool in the proofs of the results of the previous section is a minimum principle. Thisminimum principle shows in particular the relevance of (A) in our considerations. This is discussedin this section.

The following result is a variant and in fact slight generalization of the minimum principle from[21].

7


Theorem 7. (Minimum principle) Let (V, b, c) be a weighted graph and m a measure on V of fullsupport. Let U ⊆ V be connected. Assume that the function u on V satisfies

• (L+ α)u ≥ 0 on U for some α > 0,

• u ≥ 0 on V \ U .

Then, the value of u is nonnegative in any local minimum of u.

Proof. Let u attain a local minimum on U in xm. Assume u(xm) < 0. Then, u(xm) ≤ u(y) for ally ∈ U with b(xm, y) > 0. As u(y) ≥ 0 for y ∈ V \ U , we obtain u(xm)− u(y) ≤ 0 for all y ∈ Vwith b(xm, y) ≥ 0. By the super-solution assumption we find

0 ≤∑

b(xm, y)(u(xm)− u(y)) + c(xm)u(xm) +m(xm)αu(xm) ≤ 0.

As b and c are nonnegative, m is positive and α > 0, we obtain the contradiction 0 = u(xm).

The relevance of (A) comes from the following consequence of the minimum principle firstdiscussed in [21].

Proposition 8. (Uniqueness of solutions on `p) Assume (A). Let α > 0, p ∈ [1,∞) and u ∈`p(V,m) with (L + α)u ≥ 0 be given. Then, u ≥ 0. In particular, any u ∈ `p(V,m) with(L+ α)u = 0 satisfies u ≡ 0.

Proof. We first show the first statement: Assume the contrary. Then, there exists an x0 ∈ V withu(x0) < 0. By the previous minimum principle, x0 is not a local minimum of u. Thus, there existsan x1 connected to x0 with u(x1) < u(x0) < 0. Continuing in this way we obtain a sequence (xn)of connected points with u(xn) < u(x0) < 0. Combining this with (A) we obtain a contradictionto u ∈ `p(V,m).

As for the ’In particular’ part we note that both u and −u satisfy the assumptions of the firststatement. Thus, u ≡ 0.

Remark. The situation for p = ∞ is substantially more complicated as can be seen by ourdiscussion of stochastic completeness in Section 8. and in particular part (ii) of Theorem 25.

Using the previous minimum principle it is not hard to prove the following result. The result isin fact true for general Dirichlet forms as can be inferred from [28, 29]. For U ⊆ V we denote byQU the closure of the Q restricted to Cc(U) and by LU the associated operator.

Proposition 9. (Domain monotonicity) Let (V, b, c) be a symmetric graph. Let K1 ⊆ V be finiteand K2 ⊆ V with K1 ⊆ K2 be given. Then, for any x ∈ K1

(LK1 + α)−1f(x) ≤ (LK2 + α)−1f(x)

for all f ∈ `2(V,m) with f ≥ 0 and supp f ⊆ K1. A similar statement holds for the semigroups.

Proposition 10. (Convergence of resolvents/semigroups) Let (V, b, c) be a symmetric graph, ma measure on V with full support and Q the associated regular Dirichlet form. Let (Kn) be anincreasing sequence of finite subsets of V with V =

⋃Kn. Then, (LKn + α)−1f → (L + α)−1f ,

n → ∞ for any f ∈ `2(K1,mK1). (Here, (LKn + α)−1f is extended by zero to all of V .) Thecorresponding statement also holds for the semigroups.

8


3. Boundedness of the LaplacianOur main topic in this paper are the consequences of unboundedness of the Laplacian. In order tounderstand this unboundedness it is desirable to characterize boundedness of this operator. This isdiscussed in this section. We start with a little trick on how to get rid of the c in certain situations.

Let V be the union of V and a point at infinity∞. We extend a function on V to V by zero andlet b(∞, x) = b(x,∞) = c(x) for all x ∈ V . We then have

∑

y∈V

b(x, y) =∑

y∈Vb(x, y) + c(x)

for all x ∈ V andQ(u) =

1

2

∑

x,y∈V

b(x, y)(u(x)− u(y))2

for all functions u in D(Q).We define an averaged vertex degree d = db,c,m by

d(x) :=1

m(x)

(∑

y∈Vb(x, y) + c(x)

).

Note that d(x) = n(x)/m(x), where n was defined at the end of Section 1.3.

Theorem 11. Let (V, b, c) be a weighted graph and m : V −→ (0,∞) a measure on V and L theassociated formal operator. Then, the following assertions are equivalent:

(i) There exists a C ≥ 0 with d(x) ≤ C for all x ∈ V .

(ii) The form Q is bounded on `2(V,m).

(iii) The restriction of L to `2(V,m) is bounded.

(iv) The restriction of L to `∞(V ) is bounded.

In this case the restriction of L to `p(V,m) is a bounded operator for all p ∈ [1,∞] and a boundis given by 2C with C from (i).

Proof. By the considerations at the beginning of the section we can assume c ≡ 0. For x ∈ V welet δx be the function on V which is zero everywhere except in x, where it takes the value 1.

The equivalence between (ii) and (iii) is obvious as the operator associated to Q is a denselydefined restriction of L.

Obviously (i) implies (iv) (with the bound 2C). The implication (iv)=⇒ (i) follows by consid-ering the vectors δx, x ∈ V .

9


(i) =⇒ (ii): As (a− b)2 ≤ 2a2 + 2b2 we obtain

Q(u, u) =1

2

∑

x,y∈Vb(x, y)(u(x)− u(y))2

≤∑

x,y∈Vb(x, y)u(x)2 +

∑

x,y∈Vb(x, y)u(y)2

≤ C∑

x∈Vm(x)u(x)2 + C

∑

y∈Vm(y)u(y)2

= 2C‖u‖2.

Here, we used the symmetry of b and the bound (i) in the previous to the last step.(ii) =⇒ (i): This follows easily as Q(δx, δx) =

∑y∈V b(x, y) for all x ∈ V .

It remains to show the last statement: By interpolation between `2 and `∞, we obtain bound-edness of the operators on `p(V,m) for p ∈ [2,∞]. Using symmetry we obtain the boundednessfor p ∈ [1, 2). Alternatively, we can directly establish that (i) implies the boundedness of the re-striction of L on `1(V,m). As a bound for the operator norm on `∞ and on `2 is 2C, we obtain thissame bound on all `p.

Remark. The theorem can be seen as a generalization of the well known fact that a stochasticmatrix generates an operator which is bounded on all `p.

Note that the theorem gives in particular that boundedness of the operator L on `2(V,m) isequivalent to boundedness on `∞(V ). This is far from being true for all symmetric operators on`2(V,m). For example, let A be the operator on `2(N, 1) with matrix given by ax,y = 1/x if y = 1and ax,y = 1/y if x = 1 and ax,y = 0 otherwise. Then, A is bounded on `2 but not on `∞.Conversely, using e.g. the measure m(x) = x−4 on N and suitable operators with only one or twoones in each row it is not hard to construct a bounded operator on `∞(N) which is symmetric butnot bounded on `2(V,m). Of course, if m is such that `2(V,m) is contained in `∞(V ) then anybounded operator on `∞ which is symmetric (and hence closed) on `2 must be bounded as well.

4. Co-area formulaeIn this section we discuss some co-area type formulae. These formulae are well known for locallyfinite graphs e.g. [5] and carry over easily to our setting. They are useful in many contexts as e.g.the estimation of eigenvalues via isoperimetric inequalities. We use them in this spirit as well.

We start with some notation. Let (V, b, c) be a weighted graph with c ≡ 0, (which can assumewithout loss of generality by the trick mentioned in the beginning of Section 3.). For a subsetΩ ⊆ V we define

∂Ω := (x, y) : x, y ∩Ω 6= ∅ and x, y ∩ V \Ω 6= ∅

10


and|∂Ω| := 1

2

∑

(x,y)∈∂Ωb(x, y).

We can now come to the so called co-area formula.

Theorem 12. (Co-area formula) Let (V, b, c) be a weighted graph with c ≡ 0. Let f : V −→ R begiven and define for t ∈ R the set Ωt := x ∈ V : f(x) > t. Then,

1

2

∑

x,y∈Vb(x, y)|f(x)− f(y)| =

∫ ∞

0

|∂Ωt|dt.

Proof. For x, y ∈ V with x 6= y we define the interval Ix,y by

Ix,y := [minf(x), f(y),maxf(x), f(y))

and let |Ix,y| be the length of the interval. Let 1x,y be the characteristic function of Ix,y. Then,(x, y) ∈ ∂Ωt if and only if t ∈ Ix,y. Thus,

|∂Ωt| =1

2

∑

x,y∈Vb(x, y)1x,y(t).

Thus, we can calculate∫ ∞

0

|∂Ωt|dt =1

2

∫ ∞

0

∑

x,y∈Vb(x, y)1x,y(t)dt

=1

2

∑

x,y∈Vb(x, y)

∫ ∞

0

1x,y(t)dt

=1

2

∑

x,y∈Vb(x, y)|f(y)− f(x)|.

This finishes the proof.

Remark. Note that the proof is essentially a Fubini type argument.The preceding formula can be seen as a first order co-area formula as it deals with differences

of functions. There is also a zeroth order co-area type formula dealing with functions themselves.This is discussed next.

Theorem 13. Let V be a countable set and m : V −→ (0,∞) a measure on V . Let f : V −→[0,∞) be given and define for t ∈ R the set Ωt := x ∈ V : f(x) > t. Then,

∑

x∈Vm(x)f(x) =

∫ ∞

0

m(Ωt)dt.

11


Proof. We have x ∈ Ωt if and only if 1(t,∞)(f(x)) = 1. Thus, we can calculate∫ ∞

0

m(Ωt)dt =

∫ ∞

0

∑

x∈Ωt

m(x)dt

=

∫ ∞

0

∑

x∈Vm(x)1(t,∞)(f(x))dt

=∑

x∈Vm(x)

∫ ∞

0

1(t,∞)(f(x))dt

=∑

x∈Vm(x)f(x).


5. Isoperimetric inequalities and lower bounds on the (essen-tial) spectrum

In this section we will provide lower bound on the infimum of the (essential) spectrum usingan isoperimetric inequality. This will allow us in particular to provide criteria for emptiness ofthe essential spectrum. Our considerations extend the corresponding parts of [9, 11, 14, 20] (asdiscussed in more detail below).

We start with some notation used throughout this section. Let a weighted graph (V, b, c) with ameasure m : V −→ (0,∞) and the associated Dirichlet form Q be given. In this setting we definethe constant α(U) = αb,c,m(U) for a subset U ⊆ V by

α(U) = infW⊆U,|W |<∞

|∂W |m(W )

,

where as introduced in the previous section

|∂W | =∑

x∈W,y 6∈Wb(x, y) +

∑

x∈Wc(x).

Note that for a finite set W and the characteristic function 1W of W one has

|∂W |m(W )

=Q(1W )

‖1W‖2 . (5.1)

Recall the definition of the normalizing measure n on V

n(x) =∑

y∈Vb(x, y) + c(x).

12


Thus, we have two measures and thus two Hilbert spaces at our disposal. To avoid confusion, wewill write ‖ · ‖m and ‖ · ‖n for the corresponding norms whenever necessary.

Note that d(x) = n(x)/m(x). Define maximal and minimal averaged vertex degree by

dU = db,c,m(U) = infx∈U

d(x)

andDU = Db,c,m(U) = sup

x∈Ud(x),

where d is the averaged vertex degree, which was defined in Section 3. Recall d(x) = n(x)/m(x)for x ∈ V .

We will also need the restrictions of operators on V to subsets of V . As in the end of Sec-tion 2denote the closure of the restriction of a closed semibounded form Q with domain containingCc(V ) to Cc(U) by QU and its associated operator by LU (for U ⊆ V arbitrary).

For later use we also note that for the Dirichlet form Q associated to a graph (V, b, c) withmeasure m on V we have

inf σ(LU) = infu∈Cc(U)

Q(u)

‖u‖2≤ α(U) ≤ inf

x∈Ud(x) = dU

for any U ⊆ V . Here, the first equality is just the variational principle for forms, the second stepfollows from the definition of α and the last estimate follows by choosing W = x for x ∈ U .In particular, α gives upper bound on the infimum of the spectrum. It is a remarkable (and wellknown) fact that α > 0 implies also a lower bounds on the infimum of spectra. This is the core ofthe present section.

5.1. An isoperimetric inequalityIn this subsection we provide an isoperimetric inequality in our setting. This inequality (and itsproof) are generalizations of the corresponding considerations of [11, 14, 20] to our setting.

Proposition 14. Let (V, b, c) be a weighted graph, m : V −→ (0,∞) a measure on V and Q theassociated regular Dirichlet form. Let U ⊆ V and φ ∈ Cc(U). Then

Q(ϕ)2 − 2‖ϕ‖2nQ(ϕ) + αb,c,m(U)2‖ϕ‖4

m ≤ 0.

Proof. By the trick introduced at the beginning of Section we can assume without loss of generalitythat c ≡ 0. Define now A by

A =1

2

∑

x,y∈V

b(x, y)∣∣ϕ(x)2 − ϕ(y)2

∣∣ =∑

x,y∈V

b(x, y)|ϕ(x)− ϕ(y)||ϕ(x) + ϕ(y)|.

Following ideas of [11] for locally finite graphs (see [14, 20] as well) we now proceed as follows:By Cauchy-Schwarz inequality and a direct computation we have

A2 ≤ Q(ϕ)

1

2

∑

x,y∈V

b(x, y) |ϕ(x) + ϕ(y)|2 = Q(ϕ)

(2 ‖ϕ‖2

n −Q(ϕ)).

13


On the other hand we can use the first co-area formula (with f = ϕ2), the definition of α and thesecond co-area formula to estimate

A =

∫ ∞

0

|∂Ωt|dt ≥ α

∫ ∞

0

m(Ωt)dt = α∑

x∈Vm(x)ϕ2(x) = α‖ϕ‖2

m.

Combining the two estimates on A we obtain

Q(ϕ)(2 ‖ϕ‖2

n −Q(ϕ))≥ ‖ϕ‖4

m.

This yields the desired result.

5.2. Lower bounds for the infimum of the spectrumIn this section we use the isoperimetric inequality of the previous section to derive bounds on theform Q. This is in the spirit of [11, 14, 20]. As usual we write

a ≤ Q ≤ b

(for a, b ∈ R) whenevera‖u‖2 ≤ Q(u) ≤ b‖u‖2

for all u ∈ D(Q).

Proposition 15. Let (V, b, c) be a weighted graph, m : V −→ (0,∞) a measure on V and Q theassociated regular Dirichlet form. Let U ⊆ V be given and QU the restriction of Q to U . Then,

dU

(1−

√1− αb,c,n(U)2

)≤ QU ≤ DU

(1 +

√1− αb,c,n(U)2

).

If DU <∞ then furthermore

DU −√D2U − αb,c,m(U)2 ≤ QU ≤ DU +

√D2U − αb,c,m(U)2.

Proof. We start by proving the first statement. Consider an arbitrary ϕ ∈ Cc(U) with ‖ϕ‖n = 1.Then, Proposition 14 (applied with m = n) gives

Q(ϕ)2 − 2Q(ϕ) + αb,c,n(U)2 ≤ 0

and hence1−

√1− αb,c,n(U)2 ≤ Q(ϕ) ≤ 1 +

√1− αb,c,n(U)2.

As this holds for all ϕ ∈ Cc(U) with ‖ϕ‖n = 1 and

dU‖ϕ‖m ≤ ‖ϕ‖n ≤ DU‖ϕ‖m

by definition of dU and DU , we obtain the first statement.

14


We now turn to the last statement. By definition of DU we have ‖ϕ‖n ≤ DU‖ϕ‖m. Thus,Proposition 14 gives

Q(ϕ)2 − 2DU‖ϕ‖2mQ(ϕ) + αb,c,m(U)2‖ϕ‖4

m ≤ 0.

Considering now ϕ ∈ Cc(U) with ‖ϕ‖m = 1 we find that

DU −√D2U − αb,c,m(U) ≤ Q(ϕ) ≤ DU +

√D2U − αb,c,m(U)

for all such ϕ. This finishes the proof.

As a first consequence of the previous proposition we obtain the following corollary first provenfor m = n, and locally finite graphs in [14].

Corollary 16. For a weighted graph (V, b, c) and m = n we obtain

1−√

1− α2b,c,n ≤ Q ≤ 1 +

√1− α2

b,c,n.

A second consequence of the above proposition is that the bottom of the spectrum being zerocan be characterized by the constant α in the case of bounded operators. This is our version of thewell known result that a graph with finite vertex degree is amenable if and only if zero belongs tothe spectrum of the corresponding Laplacian.

Corollary 17. Let (V, b, c) be a weighted graph and DU < ∞ for U ⊆ V . Then inf σ(LU) = 0 ifand only if αb,c,m(U) = 0.

Proof. The direction ’=⇒’ follows from Proposition 15 and the other direction ’⇐=’ followsdirectly from equation 5.1.

Remark. The direction ’⇐=’ in the previous corollary does not depend on the assumption DU <∞ for U ⊆ V and is true in general.

5.3. Absence of essential spectrumIn this subsection we use the results of the previous subsection to study absence of essential spec-trum. The key idea is that the essential spectrum of an operator is a suitable limit of the spectra ofrestrictions ’going to infinity’. This reduces the problem of proving absence of essential spectrumto proving lower bounds on the spectrum ’at infinity’. For locally finite graphs this has been donein [14, 20].

Let (V, b, c) be a weighted graph. Let K be the set of finite sets in V . This set is directed withrespect to inclusion and hence a net. Limits along this net will be denoted by limK∈K and we willsay that K tends to V . We then define

αb,c,m(∂V ) = limK∈K

αb,c,m(V \K).

15


Likewise let

d∂V = db,c,m(∂V ) = limK∈K

db,c,m(V \K),

D∂V = Db,c,m(∂V ) = limK∈K

Db,c,m(V \K).

The following proposition is certainly well known and has in fact already been used in the past(see e.g. [20]). We include a proof as we could not find one in the literature. Note also that ourresult is more general than the result mentioned e.g. in [20] as we deal with forms.

Proposition 18. Let Q be a closed form on `2(V,m), whose domain of definition contains Cc(V ).If Q is bounded below then

inf σess(B) = limK∈K

inf σ(BV \K).

and if Q is bounded above then

supσess(B) = limK∈K

supσ(BV \K)

holds, where B is the operator associated to Q and BV \K the operator associated to QV \K forfinite K ⊆ V .

Proof. It suffices to show the statement for Q which are bounded below (as the other statementthen follows after replacing Q by −Q).

Without loss of generality we can assume Q ≥ 0. Let λ0 := inf σess(B).As the essential spectrum does not change by finite rank perturbations we have σess(B) =

σess(BV \K) ⊆ σ(BV \K) and henceλ0 ∈ σ(BV \K)

for any finite K ⊆ V . This gives

inf σess(B) ≥ limK∈K

inf σ(BV \K).

To show the opposite inequality it suffices to prove that for arbitrary λ < λ0 we have inf σ(BV \K) >λ for all sufficiently large finite K. Fix λ1 with

λ < λ1 < λ0

and choose δ > 0 such that λ+ δ < λ1. Moreover let

ε =λ1 − (λ+ δ)

λ1 + 1.

The spectral projectionE(−∞,λ1] ofB to the interval (−∞, λ1] is a finite rank operator sinceB ≥ 0.This easily implies

limK∈K‖E(−∞,λ1]PK‖ = 0,

16


where PK is the projection onto `2(V \K,m). Thus, there is Kε finite with

‖E(−∞,λ1]PK‖2 ≤ ε

for all K ⊇ Kε finite. In particular, we have

‖E(−∞,λ1]ψ‖2 ≤ ε (5.2)

for all ψ ∈ `2(V \Kε,m) with ‖ψ‖ = 1 (as for such ψ we have ψ = PKεψ).Consider now a finite K with K ⊇ Kε and let ψ ∈ `2(V \K,m) be given with ‖ψ‖ = 1 such

thatQ(ψ) = QV \K(ψ) ≤ (inf σ(BV \K) + ε).

Let ρψ(·) be the spectral measure associated to B and ψ. Then

Q(ψ) =

∫ ∞

0

tdρψ(t)

≥∫ ∞

λ1

t dρψ(t)

≥ λ1

∫ ∞

λ1

dρψ(t)

= λ1(〈ψ, ψ〉 − 〈E(−∞,λ1]ψ,E(−∞,λ1]ψ〉)≥ λ1(1− ε).

In the first step we used that B is positive and in the last step we used (5.2). By our choice of ψand ε we get

inf σ(BV \K) ≥ Q(ψ)− ε ≥ λ1(1− ε)− ε = λ+ δ > λ.


Combining this proposition with Proposition 15 one gets estimates for the essential spectrumof the operator L.

The following provides a generalization of a main result of Fujiwara’s theorem [14] to oursetting. Fujiwara’s result deals with locally finite graphs and m = n.

Theorem 19. Let (V, b, c) be a weighted graph, m : V −→ (0,∞) a measure on V and Q theassociated regular Dirichlet form. Assume D∂V = Db,c,m(∂V ) < ∞. Then, σess(L) = D∂V ifand only if αb,c,m(∂V ) = D∂V .

Proof. One direction ’⇐=’ follows directly from Proposition 15 and Proposition 18. The otherdirection ’=⇒’ follows from

inf σ(LU) ≤ αb,c,m(U) ≤ Db,c,m(U)

for U ⊆ V and Proposition 18 by taking U = V \K for K finite and considering the limit for Ktending to V .

17


Remark. The assumption Db,c,m(∂V ) < ∞ implies boundedness of the operator (see Sec-tion 3.). Thus, σess(L) must be non-empty in this case. Proposition 18 shows that inf σ(LV \K)and supσ(LV \K) converge necessarily to points in the essential spectrum of L (for K tendingto V ). The only way how the essential spectrum can consist of only one point is then that bothlimits agree. As inf σ(LV \K) ≤ α(V \ K) and supσ(LV \K) ≥ Db,c,m this is only possible forαb,c,m(∂V ) = D∂V . In this way the theorem characterizes the only way how essential spectrumcan consist of only one point.

The next theorem is a generalization to our setting of Theorem 2 in [20], which deals withlocally finite graphs and m ≡ 1.

Theorem 20. Let (V, b, c) be a weighted graph, m : V −→ (0,∞) a measure on V and Q theassociated regular Dirichlet form. Assume αb,c,n > 0. Then σess(L) = ∅ if and only if d∂V =∞.

Proof. One direction ’⇐=’ follows directly from Proposition 15 and 18. The other direction ’=⇒’follows from the fact that for all U ⊆ V we have inf σ(LU) ≤ db,c,m(U) and Proposition 18.

6. An applicationIn this section we consider a locally finite graph i.e., (V, b, 0) with b taking values in 0, 1 with themeasurem ≡ 1. LetQ0 be the associated form and ∆ the associated operator. Let c : V −→ [0,∞)be given and define L to be the operator associated to Qb,c,m. Thus,

L = ∆ + c

(at least on the formal level). This decomposition of L leads to a similar decomposition of theparameters α. In this way, both the geometry (encoded by b) and the potential (encoded by c) canlead to absence of essential spectrum according to the preceding considerations. This is discussedin further details next.

The Cheeger constant βU of a subset U ⊆ V is the smallest number such that for all finiteW ⊆ U

|∂W | ≥ βUvol(W ),

where |∂W | = 〈∆1W , 1W 〉 =∑

x∈W,y/∈W b(x, y) is defined as above and vol(W ) = ‖1W‖2n =∑

x∈W n(x). If βV > 0 one says that the graph is hyperbolic. Furthermore, let γU be given as thesmallest number such that for all finite W ⊆ U

c(W ) ≥ γUvol(W ),

where c(W ) = 〈c1W , 1W 〉 =∑

x∈W c(x).For example γV > 0, if there is C > 0 such that c(x) ≥ Cd(x), where d(x) is the vertex

degree.Finally let

β∂V = limK∈K

βV \K and γ∂V = limV ∈K

γV \K .

Hence the preceding section immediately gives the following corollary of Theorem 20.

18


Corollary 21. Let β∂V > 0 or γ∂V > 0. Then σess(H) = ∅ if and only if d(xn) + c(xn)→∞ alongany infinite path (xn) with pairwise distinct vertices.

7. Graph Laplacians and Markov processesWe have already discussed that our Laplacians come from Dirichlet forms. Now, Dirichlet formsand symmetric Markov processes are intimately connected. The crucial link is given by the semi-group generated by a Dirichlet form. The connection to Markov processes means that

• there is a wealth of results on the semigroup associated to a graph Laplacian,

• there is a good interpretation of properties of the semigroup in terms of a stochastic process.

Details are discussed in this section.

7.1. Graph Laplacians, their semigroup and the heat equationLet a measurem on V with full support and a graph (b, c) over V be given. LetQ be the associatedform and L its generator.

Standard theory [8, 15, 24] implies that the operators of the associated semigroup e−tL, t ≥ 0,and the associated resolvent α(L+α)−1, α > 0 are positivity preserving and even markovian. Pos-itivity preserving means that they map nonnegative functions to nonnegative functions. Markovianmeans that they map nonnegative functions bounded by one to nonnegative functions bounded byone.

This can be used to show that semigroup and resolvent extend to all `p(V,m), 1 ≤ p ≤ ∞.These extensions are consistent i.e., two of them agree on their common domain [7]. The corre-sponding generators are denoted by Lp, in particular L = L2. We can describe the action of theoperator Lp explicitly. More precisely, the situation on `2 (see Proposition 3 and Theorem 4) holdshere as well:

Theorem 22. Let (V, b, c) be a weighted graph and m a measure on V of full support. Then, theoperator Lp is a restriction of L for any p ∈ [1,∞]. If furthermore (A) holds, then the operator Lis the restriction of L to

u ∈ `p(V,m) : Lu ∈ `p(V,m).A function N : [0,∞) × V −→ R is called a solution of the heat equation if for each x ∈ V

the function t 7→ Nt(x) is continuous on [0,∞) and differentiable on (0,∞) and for each t > 0

the function Nt belongs to the domain of L, i.e., the vector space F and the equality

d

dtNt(x) = −LNt(x)

holds for all t > 0 and x ∈ V . For a bounded solution N validity of this equation can easily beseen to automatically extend to t = 0 i.e., t 7→ Nt(x) is differentiable on [0,∞) and d

dtNt(x) =

−LNt(x) holds for any t ≥ 0.

19


The following theorem is a standard result in the theory of semigroups. A proof in our contextcan be found in [21] (see [32, 33, 31] for related material on special graphs).

Theorem 23. Let L be a selfadjoint restriction of L, which is the generator of a Dirichlet formon `2(V,m). Let v be a bounded function on V and define N : [0,∞) × V −→ R by Nt(x) :=e−tLv(x). Then, the function N(x) : [0,∞) −→ R, t 7→ Nt(x), is differentiable and satisfies

d

dtNt(x) = −LNt(x)

for all x ∈ V and t ≥ 0.

Let us conclude this section by noting that the semigroups are positivity improving for con-nected graphs. This has been shown in [21] in our setting after earlier results in [8, 31, 32] forlocally finite graphs.

Theorem 24. (Positivity improving) Let (V, b, c) be a connected graph and L be the associatedoperator. Then, both the semigroup e−tL, t > 0, and the resolvent (L+ α)−1, α > 0, are positivityimproving (i.e., they map nonnegative nontrivial `2-functions to strictly positive functions).

7.2. Connection to Markov processesIn this section we discuss the relationship between Dirichlet forms and Markov processes in ourcontext. Let Q be the Dirichlet form associated to a weighted graph (V, b, c) with measure m.For convenience we assume m ≡ 1. Let L be the associated operator and e−tL, t > 0, theassociated semigroup. We will take the point of view that we already know that e−tL is a semigroupof transition properties of a Markov process. We will then show how we can identify the keyquantities of the Markov process in terms of the graph (V, b, c).

A (time homogenous) Markov process on V consists of a particle moving in time withoutmemory between the points of V . It is characerized by two sets of quantities: These are

• a function a : V −→ [0,∞) such that e−tax is the probability that a particle in x at time 0 isstill in x at time t.

• a function q : V × V −→ [0,∞) such that qx(y) is the probability that the particle jumps toy from x.

Given such a Markov process we can define

Pt(x, y) := Probability that the particle is in y at time t if it starts in x at time 0

for t ≥ 0, x, y ∈ V and the operators Pt provide a semigroup of operators. It is then possible toinfer the quantities a and q from the behavior of Pt for small t in the following way:

Pt(x, x) is the probability to find the particle at x at time t (for a particle starting at x at time 0).This means that the particle has either stayed at x for the whole time between 0 and t or has jumped

20


from x away and come back by the time t. The probability that the particle stayed in x (i.e., didnot move away) is e−tax . The event that the particle left x and returned by the time t means that theparticle left x, which occurs with probability 1− e−tax , and then returned from V \ x to x in theremaining time, which occurs with probability r(t) going to zero for t→ 0. Accordingly we have

Pt(x, x) = e−tax + φx(t),

where φx summarizes the probability of returning to x, is therefore bounded by (1−e−tax)r(t) andhence has derivative equal to zero at t = 0. We therefore obtain

d

dt

∣∣∣∣t=0

Pt(x, x) = −ax + φ′x(0) = −ax.

By a similar reasoning the probability Pt(x, y) is governed by the event that the particle starts at xat time 0 and has done one jump to y and then stayed in y up to the time t. The probability pt forthis event satisfies

(1− e−tax)qx(y)e−tay ≤ pt ≤ (1− e−tax)qx(y).

Here, the term e−tay serves to take into account that the particle did not leave y. Accordingly,

Pt(x, y) = pt + ψ(t),

where the derivative of ψ at 0 is zero and we obtain

d

dt

∣∣∣∣t=0

Pt(x, y) = axqx(y) + ψ′(0) = axqx(y).

We now return to the Dirichlet form setting. As e−tL describes a Markov process we can nowset

Pt(x, y) = 〈e−tLδx, δy〉for t ≥ 0, x, y ∈ V and use this to calculate the the a’s and q’s in terms of b and c as follows:

∑

y∈Vb(x, y) + c(x) = Q(δx, δx) =

d

dt

∣∣∣∣t=0

〈e−tLδx, δx〉 =d

dt

∣∣∣∣t=0

Pt(x, x) = −ax

and

−b(x, y) = Q(δx, δy) =d

dt

∣∣∣∣t=0

〈e−tLδx, δy〉 =d

dt

∣∣∣∣t=0

Pt(x, y) = qx(y)ax.

This gives

qx(y) =b(x, y)∑

z∈V b(x, z) + c(x), ax =

∑

z∈Vb(x, z) + c(x)

for all x, y ∈ V . Note that symmetry of b does not imply symmetry of q but rather

axqx(y) = ayqy(x).

If m is not identically equal to one, we will have to normalize the formula for P above by setting

Pt(x, y) =1

m(x)m(y)〈e−tLδx, δy〉

and change the emerging formulae accordingly.

21


8. Stochastic completenessWe consider a Dirichlet form Q on a weighted graph (V, b, c) with associated operator L andsemigroup e−tL. The preceding considerations show that

0 ≤ e−tL1(x) ≤ 1

for all t ≥ 0 and x ∈ V . The question, whether the second inequality is actually an equality hasreceived quite some attention. In the case of vanishing killing term, this is discussed under thename of stochastic completeness or conservativeness. In fact, for c ≡ 0 and b(x, y) ∈ 0, 1 forall x, y ∈ V , there is a characterization of stochastic completeness of Wojciechowski [32] (see theintroduction for discussion of related results of Feller [13] and Reuter [26] as well). This charac-terization is an analogue to corresponding results on manifolds of Grigor’yan [16] and results ofSturm for general strongly local Dirichlet forms [27].

Our first main result concerns a version of this result for arbitrary regular Dirichlet forms ongraphs. As we allow for a killing term c we have to replace e−tL1 by the function

Mt(x) := e−tL1(x) +

∫ t

0

e−sLc(x)ds, x ∈ V.

It is possible (and necessary) to show that this quantity is well defined. In fact, it can be proventhat it satisfies 0 ≤ M ≤ 1 and that for each x ∈ V , the function t 7→ Mt(x) is continuous andeven differentiable [21].

Note that for c ≡ 0, M = e−tL1 whereas for c 6= 0 the inequality Mt > e−tL1 holds on anyconnected component of V on which c does not vanish identically (as the semigroup is positivityimproving).

We can give an interpretation of M in terms of a diffusion process on V as follows: For x ∈ V ,let δx be the characteristic function of x. A diffusion on V starting in xwith normalized measureis then given by 1

m(x)δx at time t = 0. It will yield to the amount of heat

〈e−tL δxm(x)

, 1〉 = 〈 δxm(x)

, e−tL1〉 =∑

y∈Ve−tL(x, y) = e−tL1(x)

within V at the time t. Thus, the first term of M describes the amount of heat within the graph ata given time.

Moreover, at each time s the rate of heat killed at the vertex y by the killing term c is given byc(y)e−sL(x, y). The total amount of heat killed at y till the time t is then given by

∫ t0c(y)e−sL(x, y)ds.

The total amount of heat killed at all vertices by c till the time t is accordingly given by

∑

y∈V

∫ t

0

c(y)e−sL(x, y)ds =

∫ t

0

∑

y∈Ve−sL(x, y)c(y)ds =

∫ t

0

(e−sLc)(x)ds.

Thus, the second term of M describes the total amount of heat killed up to time t within the graph.Altogether, 1−Mt is then the amount of heat transported to the ’boundary’ of the graph by the time

22


t and Mt can be interpreted as the amount of heat, which has not been transported to the boundaryof the graph at time t.

Our question concerning stochastic completeness then becomes whether the quantity

1−Mt

vanishes identically or not. Our result reads (see [21] for a proof):

Theorem 25. (Characterization of heat transfer to the boundary) Let (V, b, c) be a weighted graphand m a measure on V of full support. Then, for any α > 0, the function

w :=

∫ ∞

0

αe−tα(1−Mt)dt

satisfies 0 ≤ w ≤ 1, solves (L + α)w = 0, and is the largest nonnegative function l ≤ 1 with(L+ α)l ≤ 0. In particular, the following assertions are equivalent:

(i) For any α > 0 there exists a nontrivial, nonnegative, bounded l with (L+ α)l ≤ 0.

(ii) For any α > 0 there exists a nontrivial, bounded l with (L+ α)l = 0.

(iii) For any α > 0 there exists an nontrivial, nonnegative, bounded l with (L+ α)l = 0.

(iv) The function w is nontrivial.

(v) Mt(x) < 1 for some x ∈ V and some t > 0.

(vi) There exists a nontrivial, bounded, nonnegative N : V × [0,∞) −→ [0,∞) satisfyingLN + d

dtN = 0 and N0 ≡ 0.

Let us give a short interpretation of the conditions appearing in the theorem. Conditions (i),(ii) and (iii) deal with eigenvalues of L considered as an operator on `∞(V ). Thus, they concernspectral theory in `∞(V ). Condition (v) refers to loss of mass at infinity. Finally condition (vi) isabout unique solutions of a partial difference equation. Thus, the result connects properties fromstochastic processes, spectral theory and partial difference equations.

Sketch of proof. We refrain from giving a a complete proof of the theorem but rather discuss threekey elements of the proof and how they fit together. These are the following three steps:

(S1) If N : [0,∞)× V −→ R is a bounded solution of ddtN = −LN , then v =

∫∞0αe−tαNtdt is

a solution to (L+ α)v = 0 for any α > 0.

(S2) The function N = 1−M satisfies 0 ≤ N ≤ 1 and ddtN = −LN .

(S3) The function w =∫∞

0αe−tα(1 −Mt)dt is the largest solution of (L + α)v = 0 with 0 ≤

v ≤ 1.

23


The proof of the first step is a direct calculation via partial integration. The second step is adirect calculation but requires quite some care as the quantities are defined via sums and integralswhose convergence is not clear. The fact thatw of the last step is a solution follows from the secondstep. The minimality of the solution requires some care. It follows by approximating the graph viafinite graphs. Here, a nontrivial issue is that this approximation may actually cut infinitely manyedges (as we do not have locally finite edge degree).

Given the three steps, the proof of the theorem goes along the following line: The implication(v) =⇒ (i) follows from Step (S1) and (S2). The implication (i) =⇒ (v) follows from the maxi-mality property in Step (3). The implication (v) =⇒ (vi) follows from Step (S2). The implication(vi) =⇒ (v) follows from Step (S1). The equivalence between (iv) and (v) is immediate fromStep (S3). The equivalence between (i), (ii) and (iii) follows by taking suitable minima of (super-)solutions.

Definition 26. The weighted graph (V, b, c) is said to be stochastically complete if one of theequivalent assertions of the theorem holds.

Corollary 27. Assume the situation of the previous theorem. Let L be the operator associatedto the graph (V, b, c). If L gives rise to a bounded operator on `2(V ), then the graph (V, b, c) isstochastically complete.

Proof. If L is bounded on `2(V,m) it is bounded on `∞(V ) by Theorem 11. Then, the spectrum ofL on `∞ is bounded and hence its set of eigenvalues is bounded as well. Thus, (ii) of the theoremmust fail (for large α).

Remark. (a) The corollary shows that stochastic completeness is a phenomenon for unboundedoperators.

(b) The corollary generalizes the results of Dodziuk/Matthai [12] and Wojciechowski [32]. Itis furthermore relevant as its proof gives an abstract i.e., spectral theoretic reason for stochasticcompleteness in the case of bounded operators.

Let us finish this section by discussing how the existence of α > 0 and t > 0 and x ∈ V withcertain properties in the above theorem is actually equivalent to the fact that all α > 0 , t > 0 andx ∈ V have these properties. We first discuss the situation concerning the α’s.

Proposition 28. Let (V, b, c) be a weighted graph and m a measure on V of full support. Then,the following are equivalent:

(i) For any α > 0 there exists a nontrivial, nonnegative, bounded l with (L+ α)l ≤ 0.

(ii) For some α > 0 there exists a nontrivial, nonnegative, bounded l with (L+ α)l ≤ 0.

Proof. It suffices to show the implication (ii) =⇒ (i): By the maximality property of the functionw =

∫∞0αe−tα(1 −Mt)dt discussed in the third step of the proof of the main result, (ii) implies

that Mt(x) < 1 for some x ∈ V and t > 0. Now, the claim (i) follows from the second stepdiscussed in the proof of the main result.

24


We now show that loss of mass in one point at one time is equivalent to loss of mass in allpoints at all times (if the graph is connected). For locally finite graphs this is discussed in [32].

Proposition 29. Let (V, b, c) be a connected weighted graph andm a measure on V of full support.Let M be defined as above. Then, the following assertions are equivalent:

(i) There exist x ∈ V and t > 0 with Mt(x) < 1.

(ii) Mt(x) < 1 for all x ∈ V and all t > 0

Proof. The implication (ii) =⇒ (i) is clear. It remains to show the reverse implication. A directcalculation (invoking

∫ t+s0

...dr =∫ s

0...dr +

∫ t+ss

...dr) shows that

Mt+s = e−sLMt +

∫ s

0

e−rLc dr.

This easily gives that

(1) Mt ≡ 1 for some t > 0 implies Mnt ≡ 1 for all n ∈ N.

(2) Mt 6= 1 for some t > 0 implies that Mt+s < 1 for all s > 0.

(Here (1) follows by induction and (2) follows as Mt 6= 1 implies Mt ≤ 1 and Mt(x) < 1 forsome x ∈ V . As the graph is connected this implies e−sLMt < esL1 and the statement follows.)

Assume now that Mt(x) < 1 for some x ∈ V and t > 0. We consider Mr for r > t and forr < t separately: By (2), Mr < 1 for all r > t. Assume that Mr = 1 for some 0 < r < t, thenMs = 1 for all s ≤ r by (2). Hence, by (1) Mns = 1 for all n ∈ N and 0 < s ≤ r. This givesMr = 1 for all r > 0 which contradicts Mt 6= 1. Thus, Mr 6= 1 for all 0 < r < t. Hence, by (2)Mr < 1 for all 0 < r ≤ t.

Acknowledgements. It is our great pleasure to acknowledge fruitful and stimulating discus-sions with Peter Stollmann, Radek Wojciechowski, Andreas Weber and Jozef Dodziuk on thetopics discussed in the paper.

References[1] A. Beurling, J. Deny. Espaces de Dirichlet. I. Le cas elementaire. Acta Math., 99 (1958),

203–224.

[2] A. Beurling, J. Deny. Dirichlet spaces. Proc. Nat. Acad. Sci. U.S.A., 45 (1959), 208–215.

[3] N. Bouleau, F. Hirsch. Dirichlet forms and analysis on Wiener space. Volume 14 of deGruyter Studies in Mathematics, Walter de Gruyter & Co., Berlin, 1991.

[4] F. R. K. Chung. Spectral Graph Theory. CBMS Regional Conference Series in Mathematics,92, American Mathematical Society, Providence, RI, 1997.

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[5] F. R. K. Chung, A. Grigoryan, S.-T. Yau. Higher eigenvalues and isoperimetric inequalitieson Riemannian manifolds and graphs. Comm. Anal. Geom., 8 (2000), No. 5, 969–1026.

[6] Y. Colin de Verdiere. Spectres de graphes. Soc. Math. France, Paris, 1998.

[7] E. B. Davies. Heat kernels and spectral theory. Cambridge University press, Cambridge,1989.

[8] E. B. Davies. Linear operators and their spectra. Cambridge Studies in Advanced Mathemat-ics, 106. Cambridge University Press, Cambridge, 2007.

[9] J. Dodziuk. Difference Equations, isoperimetric inequality and transience of certain randomwalks. Trans. Amer. Math. Soc., 284 (1984), No. 2, 787–794.

[10] J. Dodziuk. Elliptic operators on infinite graphs. Analysis, geometry and topology of ellipticoperators, 353–368, World Sci. Publ., Hackensack, NJ, 2006.

[11] J. Dodziuk, W. S. Kendall. Combinatorial Laplacians and isoperimetric inequality. Fromlocal times to global geometry, control and physics (Coventry, 1984/85), 68–74, Pitman Res.Notes Math. Ser., 150, Longman Sci. Tech., Harlow, 1986.

[12] J. Dodziuk, V. Matthai. Kato’s inequality and asymptotic spectral properties for discretemagnetic Laplacians. The ubiquitous heat kernel, 69–81, Contemp. Math., 398, Amer. Math.Soc., Providence, RI, 2006.

[13] W. Feller. On boundaries and lateral conditions for the Kolmogorov differential equations.Ann. of Math. (2), 65 (1957), 527–570.

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[20] M. Keller. The essential spectrum of Laplacians on rapidly branching tesselations. Math.Ann., 346 (2010), No. 1, 51–66.

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[23] B. Mohar. Light structures in infinite planar graphs without the strong isoperimetric property.Trans. Amer. Math. Soc., 354 (2002), No. 8, 3059–3074.

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[25] B. Metzger, P. Stollmann. Heat kernel estimates on weighted graphs. Bull. London Math.Soc., 32 (2000), No. 4, 477–483.

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[27] K.-T. Sturm. textitAnalysis on local Dirichlet spaces. I: Recurrence, conservativeness andLp-Liouville properties. J. Reine Angew. Math., 456 (1994), No. 173–196.

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27

CHAPTER 6

M. Keller, D. Lenz, R. Wojciechowski, Volumegrowth, spectrum and stochastic completeness ofinfinite graphs, Mathematische Zeitschrift 274

(2013), 905–932.

139

Math. Z. (2013) 274:905–932DOI 10.1007/s00209-012-1101-1 Mathematische Zeitschrift

Volume growth, spectrum and stochastic completenessof infinite graphs

Matthias Keller · Daniel Lenz ·Radosław K. Wojciechowski

Received: 16 February 2012 / Accepted: 16 September 2012 / Published online: 16 November 2012© Springer-Verlag Berlin Heidelberg 2012

Abstract We study the connections between volume growth, spectral properties and sto-chastic completeness of locally finite weighted graphs. For a class of graphs with a very weakspherical symmetry we give a condition which implies both stochastic incompleteness anddiscreteness of the spectrum. We then use these graphs to give some comparison results forboth stochastic completeness and estimates on the bottom of the spectrum for general locallyfinite weighted graphs.

Mathematics Subject Classification (2000) Primary 39A12; Secondary 58J35

1 Introduction

The aim of this paper is to investigate the connections between volume growth, uniqueness ofbounded solutions for the heat equation, and spectral properties for infinite weighted graphs.To do so, we proceed in two steps. We first establish these connections on a class of graphswith a very weak spherical symmetry. We then give some comparison results for generalgraphs by using a notion of curvature.

A very general framework for studying operators on discrete measure spaces was recentlyestablished in [37]. We use this set up throughout with the additional assumption that theunderlying weighted graphs are locally finite.

M. Keller · D. LenzMathematisches Institut, Friedrich Schiller Universität Jena,07743 Jena, Germanye-mail: [email protected]

D. Lenze-mail: [email protected]

R. K. Wojciechowski (B)York College of the City University of New York, Jamaica,NY 11451, USAe-mail: [email protected]

123

906 M. Keller et al.

We first introduce the class of weakly spherically symmetric graphs which, comparedwith spherically symmetric graphs, can have very little symmetry. Still, these graphs turn outto be accessible to a detailed analysis much as those with a full spherical symmetry. Moreprecisely, we can characterize them by the fact that:

• Their heat kernels are spherically symmetric (Theorem 1 in Sect. 3).

Moreover, for operators arising on these graphs we prove:

• An explicit estimate for the bottom of their spectrum and a criterion for the discretenessof the spectrum in terms of volume growth and the boundary of balls (Theorem 3 inSect. 4).

• A characterization of stochastic completeness in terms of volume growth and the bound-ary of balls (Theorem 5 in Sect. 5).

Both the estimate for the bottom of the spectrum and the condition for stochastic completenessinvolve the ratio of a generalized volume of a ball to its weighted boundary. In this sense, ourestimates complement the classic lower bound on the bottom of the spectrum of the standardgraph Laplacian in terms of Cheeger’s constant given by Dodziuk in [15], in the case ofunbounded geometry.

These results give rise to examples of graphs of polynomial volume growth which havepositive bottom of the spectrum and are stochastically incomplete. Therefore, as a surprisingconsequence, in the standard graph metric there are no direct analogues to the theorems ofGrigor’yan, relating volume growth and stochastic completeness [25], and of Brooks, relatingvolume growth and the bottom of the essential spectrum [9], from the manifold setting. Forstochastic completeness this was already observed in [57]. These examples are studied at theend of the paper, in Sect. 6.

We now turn to the second step of our investigation, i.e., the comparison of generalweighted graphs to weakly spherically symmetric ones. In this context, we provide:

• heat kernel comparisons (Theorem 2 in Sect. 3)• comparisons for the bottom of the spectrum (Theorem 4 in Sect. 4)• comparisons for stochastic completeness (Theorem 6 in Sect. 5).

The heat kernel comparison, Theorem 2, is given in the spirit of results of Cheeger and Yau[12]. However, in contrast to other works done for graphs in this area, e.g. [51,56], as we useweakly spherically symmetric graphs, we require very little symmetry for our comparisonspaces. The comparisons are then done with respect to a certain curvature (and, in the caseof stochastic completeness, also to potential) growth. We combine these inequalities with ananalogue to a theorem of Li [41], which was recently proven in our setting in [28,39], toobtain comparisons for the bottom of the spectrum, Theorem 4. The spectral comparisonsgive some analogues to results of Cheng [13] and extend inequalities found for graphs in[10,52,58]. The comparison results for stochastic completeness, Theorem 6, are inspired bywork of Ichihara [34] and are found in Sect. 5.

The article is organized as follows: in the next section we introduce the set up. This isfollowed by the heat kernel theorems, Theorems 1 and 2, in Sect. 3, the spectral estimates,Theorems 3 and 4, in Sect. 4 and the considerations about stochastic completeness, Theo-rems 5 and 6, in Sect. 5. The proofs are given within each section. In Sect. 6, we discussthe applications to standard graph Laplacians and give the examples of polynomial volumegrowth announced above. Finally, in Appendix A, we prove some general facts concerningcommuting operators which are used in the proof of Theorem 1.

123

Volume growth, spectrum and stochastic completeness 907

2 The set up and basic facts

Our basic set up, which is included in [37,38], is as follows: let V be a countably infiniteset and m : V → (0,∞). Extending m to all subsets of V by countable additivity gives ameasure of full support and (V, m) is then a measure space. The map b : V × V → [0,∞),which characterizes the edges, is symmetric and has zero diagonal. If b(x, y) > 0, then wesay that x and y are neighbors, writing x ∼ y, and think of b(x, y) as the weight of the edgeconnecting x and y. Moreover, let c : V → [0,∞) be a map which we call the potential orkilling term. If c(x) > 0, then we think of x as being connected to an imaginary vertex atinfinity with weight c(x). We call the quadruple (V, b, c, m) a weighted graph. Wheneverc ≡ 0, we denote the weighted graph (V, b, 0, m) as the triple (V, b, m). If, furthermore,b : V × V → 0, 1, then we speak of (V, b, m) as a standard graph.

We say that a weighted graph is connected if, for any two vertices x and y, there exists asequence of vertices (xi )

ni=0 such that x0 = x , xn = y and xi ∼ xi+1 for i = 0, 1, . . . , n −1.

We say that a weighted graph is locally finite if every x ∈ V has only finitely many neighbors,i.e., b(x, y) vanishes for all but finitely many y ∈ V .

Throughout the paper we assume that all weighted graphs (V, b, c, m) in question areconnected and locally finite.

In this setting, weighted graph Laplacians and Dirichlet forms on discrete measure spaceswere recently studied in [37], whose notation we closely follow (see also the seminal work[6] on finite graphs and [24] for background on general Dirichlet forms). Let C(V ) denotethe set of all functions from V to R and let

2(V, m) =

f ∈ C(V ) |∑x∈V

f 2(x)m(x) < ∞

denote the Hilbert space of functions square summable with respect to m with inner productgiven by

〈 f, g〉 =∑x∈V

f (x)g(x)m(x).

We then define the form Q with domain of definition

D(Q) = Cc(V )〈·,·〉Q

,

where Cc(V ) denotes the space of finitely supported functions, the closure is taken withrespect to 〈·, ·〉Q := 〈·, ·〉 + Q(·, ·) in 2(V, m), and Q acts by

Q( f, g) = 1

2

∑x,y∈V

b(x, y) ( f (x) − f (y)) (g(x) − g(y)) +∑x∈V

c(x) f (x)g(x).

Such forms are regular Dirichlet forms on the measure space (V, m), see [37].By general theory (see, for instance [24]) there is a selfadjoint positive operator L with

domain D(L) ⊆ 2(V, m) such that Q( f, g) = 〈L f, g〉 for f ∈ D(L) and g ∈ D(Q).By [37, Theorem 9] we know that L is a restriction of the formal Laplacian L which actsas

L f (x) = 1

m(x)

∑y∈V

b(x, y) ( f (x) − f (y)) + c(x)

m(x)f (x)

123


and, as (V, b, c, m) is locally finite, is defined for all functions in C(V ). Alternatively, asshown in [37] as well, one can consider L to be an extension—the so-called Friedrichsextension—of the operator L0 defined on Cc(V ) by

L0g = Lg.

Two very prominent examples from the standard setting are the graph Laplacian givenby the additional assumptions that b : V × V → 0, 1, c ≡ 0, and m ≡ 1 so that

f (x) =∑y∼x

( f (x) − f (y))

and the normalized graph Laplacian given by the additional assumptions that b : V ×V →0, 1, c ≡ 0, and m ≡ deg so that

f (x) = 1

deg(x)

∑y∼x

( f (x) − f (y)) .

Here, deg(x) = |y | y ∼ x| for x ∈ V , where |·| denotes the cardinality of a set, is finitefor all x ∈ V by the local finiteness assumption.

In the following sections, we will compare our results to those for and found in theliterature. Furthermore, we will illustrate some of our results for the operator at the end ofthe paper in Sect. 6. We note that is bounded on 2(V ) := 2(V, 1) if and only if deg isbounded, while is always bounded on 2(V, deg).

We will often fix a vertex x0 and consider spheres and balls

Sr = Sr (x0) = x | d(x, x0) = r and Br = Br (x0) =r⋃

i=0

Si (x0)

around x0 of radius r . Here, d(x, y) is the usual combinatorial metric on graphs, that is, thenumber of edges in the shortest path connecting x and y.

Definition 2.1 The outer and inner curvatures κ± : V → [0,∞) are given by

κ±(x) = 1

m(x)

∑y∈Sr±1

b(x, y) for x ∈ Sr .

Remark We refer to these quantities as curvatures as, for c ≡ 0, Ld(x0, ·) = κ−(·) − κ+(·)is often referred to as a curvature-type quantity for graphs, see [17,33,54]. Moreover, in lightof our Theorem 1 and [12, Proposition 2.2], as well as our comparison results, Theorems 2,4, and 6, and their counterparts in the manifold setting which can be found in [12,13,34], itseems reasonable to relate κ± to a type of curvature on a manifold.

Several other notions of curvature have been introduced for planar graphs [4,5,31], cellcomplexes [21], and general metric measure spaces [8,43,45,49,50]. See also the recent workon Ricci curvature of graphs [3,35,42]. It is not presently clear how these different notionsof curvature are related.

Definition 2.2 We call a function f : V → R spherically symmetric if its values dependonly on the distance to x0, i.e., if f (x) = g(r) for x ∈ Sr (x0) for some function g definedon N0 = 0, 1, 2, . . .. In this case, we will often write f (r) for f (x) whenever x ∈ Sr (x0)

and set, for convenience, f (−1) = 0.

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Let A be the operator on C(V ) that averages a function over a sphere around x0, i.e.,

( A f )(x) = 1

m(Sr )

∑y∈Sr

f (y)m(y)

for f : V → R and x ∈ Sr . This operator is a projection whose range is the sphericallysymmetric functions. In particular, a function f is spherically symmetric if and only ifA f = f . Moreover, the restriction A of A to 2(V, m) is bounded and symmetric. Therefore,A is an orthogonal projection.

Definition 2.3 Let the normalized potential q : V → [0,∞) be given by

q = c

m.

We call a weighted graph (V, b, c, m) weakly spherically symmetric if it contains a vertexx0 such that κ± and q are spherically symmetric functions.

We call the vertex x0 in the definition above the root of (V, b, c, m).

Remark We will often suppress the dependence on the vertex x0. Mostly, we will denoteweakly spherically symmetric graphs by (V sym, bsym, csym, msym) although bsym, csym andmsym might not have any obvious symmetries at all. Furthermore, we will denote, as needed,the corresponding curvatures and potential by κ

sym± and qsym and the Laplacian on such a

graph by Lsym.

For a weakly spherically symmetric graph the operator L acts on a spherically symmetricfunction f by

L f (r) = κ+(r)( f (r) − f (r + 1)) + κ−(r)( f (r) − f (r − 1)) + q(r) f (r).

Moreover, a straightforward calculation yields that

κ+(r)m(Sr ) = κ−(r + 1)m(Sr+1) for all r ∈ N0. (2.1)

Let us give some examples to illustrate the definition of weakly spherically symmetricgraphs.

Example (a) We call a weighted graph (V, b, c, m) spherically symmetric with respect tox0 if for each x, y ∈ Sr (x0), r ∈ N0, there exists a weighted graph automorphism whichleaves x0 invariant and maps x to y. In this case, the weighted graph is weakly sphericallysymmetric.

(b) If the functions k± := κ±m, the potential c and the measure m are all sphericallysymmetric functions, then the weighted graph is weakly spherically symmetric. On theother hand, given that the measure m is spherically symmetric and the graph is weaklyspherically symmetric, then k± and c must be spherically symmetric.

Remark The second example shows that there are very little assumptions on the symmetryof the geometry in the weakly spherically symmetric case. The difference is illustrated inFig. 1. There, standard graphs, that is, with c ≡ 0 and b : V × V → 0, 1, and constantmeasure are plotted up to the third sphere. The first and second graphs are only weaklyspherically symmetric, while the third one is spherically symmetric. However, given the lackof assumptions on connections within a sphere and the structure of connections between thespheres, the freedom in the weakly spherically symmetric case is even much greater thanillustrated in the figure. Indeed, we do not have any assumptions on the vertex degree as longas the outer and inner curvatures are constant on each sphere.

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Fig. 1 The first two standard graphs are only weakly spherically symmetric while the third one is sphericallysymmetric

3 Heat kernel comparisons

3.1 Notations and definitions

Let a weighted graph (V, b, c, m) be given. For the operator L acting on D(L) ⊆ 2(V, m)

we know, by the discreteness of the underlying space, that there exists a map

p : [0,∞) × V × V → R,

which we call the heat kernel associated to L , with

e−t L f (x) =∑y∈V

pt (x, y) f (y)m(y)

for all f ∈ 2(V, m). Here, e−t L is the operator semigroup of L which is defined via thespectral theorem. By direct computation, one sees that pt (x, y) = e−t Lδy(x), for x, y ∈ Vand t ≥ 0, where δy is a 1-normalized delta function, i.e., δy(x) = 1

m(y)if x = y and zero

otherwise.

Definition 3.1 We say that the heat kernel p of an operator L is spherically symmetric ifthere is a vertex x0 such that the averaging operator A and the semigroup e−t L commute forall t ≥ 0.

In particular, in this case, the function pt (x0, ·) is spherically symmetric for each t and,whenever this is the case, we write

psymt (r) = pt (x0, x)

for x ∈ Sr and r ∈ N0.In order to compare a general weighted graph with a weakly spherically symmetric one,

we introduce the following terminology.

Definition 3.2 A weighted graph (V, b, c, m) has stronger (respectively, weaker) cur-vature growth with respect to x0 ∈ V than a weakly spherically symmetric graph(V sym, bsym, csym, msym) with root o if, m(x0) = msym(o) and if, for all r ∈ N0 andx ∈ Sr ⊂ V ,

κ+(x) ≥ κsym+ (r) and κ−(x) ≤ κ

sym− (r)

(respectively, κ+(x) ≤ κsym+ (r) and κ−(x) ≥ κ

sym− (r)).

Remark The assumption m(x0) = msym(o) is a normalization condition, necessary for ourcomparison theorems below, which states that the same amount of heat enters both graphs atthe root. This follows since, in general, p0(x, y) = 1

m(x)if x = y and 0 otherwise.

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3.2 Theorems and remarks

There are two main results about heat kernels which are proven in this section. The first one,which is an analogue for Proposition 2.2 in [12] from the manifold setting, is that weaklyspherically symmetric graphs can be characterized by the symmetry of the heat kernel.

Theorem 1 (Spherical symmetry of heat kernels) A weighted graph (V, b, c, m) is weaklyspherically symmetric if and only if the heat kernel is spherically symmetric.

Remark It is clear that the heat kernel on a spherically symmetric graph is spherically sym-metric. However, the theorem also implies that the heat kernels of all the graphs illustratedin Fig. 1 are spherically symmetric.

The second main result of this section is a heat kernel comparison between weakly spher-ically symmetric graphs and general weighted graphs. These comparisons were originallyinspired by [12] and can be found in [56] for the graph Laplacian and trees. See also [51]for related results in the case of the graph Laplacian, regular trees, and heat kernels with adiscrete time parameter.

Theorem 2 (Heat kernel comparison with weakly spherically symmetric graphs) If aweighted graph (V, b, m) with heat kernel p has stronger (respectively, weaker) curva-ture growth than a weakly spherically symmetric graph (V sym, bsym, msym) with heat kernelpsym, then for x ∈ Sr (x0) ⊂ V , r ∈ N0, and t ≥ 0,

psymt (r) ≥ pt (x0, x)

(respectively, psym

t (r) ≤ pt (x0, x)).

3.3 Proofs of Theorems 1 and 2

We start by developing the ideas necessary for the proof of Theorem 1. We first characterizethe class of weakly spherically symmetric graphs using A, the averaging operator, in thefollowing way.

Lemma 3.3 Let (V, b, c, m) be a weighted graph and x0 ∈ V . Let A be the averagingoperator associated to x0. Then, the following assertions are equivalent:

(i) (V, b, c, m) is weakly spherically symmetric, i.e., κ± and q are spherically symmetricfunctions.

(ii) L commutes with A, i.e., AL f = L A f for all f ∈ C(V ).(iii) L0 commutes with A on Cc(V ), i.e., AL0g = L0 Ag for all g ∈ Cc(V ).

Proof The direction (i) ⇒ (ii) follows by straightforward computation using the formulasκ+(r)m(Sr ) = κ−(r + 1)m(Sr+1), see (2.1).

As A and L are restrictions of A and L , their matrix elements agree. This gives the direction(ii) ⇒ (iii).

We finally turn to (iii) ⇒ (i). The function 1Sr , which is one on Sr and zero elsewhere,satisfies

L0 A1Sr (x) = L01Sr (x) =⎧⎨⎩

κ+(x) + κ−(x) + q(x) if x ∈ Sr

−κ∓(x) if x ∈ Sr±1

0 otherwise

and

AL01Sr (x) =

⎧⎪⎨⎪⎩

1m(Sr )

∑y∈Sr

(κ+(y) + κ−(y) + q(y))m(y) if x ∈ Sr

− 1m(Sr±1)

∑y∈Sr±1

κ∓(y)m(y) if x ∈ Sr±1

0 otherwise.

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By (iii), these two expressions must be equal which yields that κ± and q must be sphericallysymmetric functions which is (i).

From Corollary A.8 in Appendix A with H = 2(V, m) and D0 = Cc(V ) we can nowinfer the following statement. Recall that A is the restriction of A to 2(V, m) which is anorthogonal projection onto the subspace of spherically symmetric functions.

Lemma 3.4 Let (V, b, c, m) be a weighted graph. Then, the following assertions are equiv-alent:

(i) AL0 f = L0 A f for all f ∈ Cc(V ).(ii) A maps D(L) into D(L) and AL f = L A f for all f ∈ D(L).

(iii) e−t L commutes with A for all t ≥ 0.

Remark A proof of the implication (i) ⇒ (ii) in Lemma 3.4 could also be based on Propo-sition A.4 from the appendix and the approximation of the heat kernel p by the restrictedkernels pi on Hi := 2(Bi , mi ) discussed below.

Proof of Theorem 1 To prove Theorem 1 simply combine Lemmas 3.3 and 3.4. The following construction of the heat kernel, first presented in the continuous setting in

[14] and carried over to our set-up in [37], will be crucial. Let x0 ∈ V and Bi = Bi (x0)

denote the corresponding distance balls. By the connectedness of the graph, V = ⋃∞i=0 Bi

and, as Bi ⊆ Bi+1, the balls are an increasing exhaustion sequence of the graph (V, b, c, m).For i ∈ N0, let mi be the restriction of m to Bi and consider the restriction L(D)

i of theLaplacian L to the finite dimensional space 2(Bi , mi ) with Dirichlet boundary conditions.This operator can be defined by restricting the form Q to Cc(Bi ) and taking the closure in2(Bi , mi ) with respect to 〈·, ·〉Q . It turns out, (see [37]), that

L(D)i f (x) = L f (x) for f ∈ 2(Bi , mi ) and x ∈ Bi .

This just means that L(D)i = πi Lιi for the canonical injection ιi : 2(Bi , mi ) −→ 2(V, m)

acting as extension by zero and πi , the adjoint of ιi .We let pi denote the heat kernel of L(D)

i which is extended to V × V by zero. That is,

pit (x, y) =

e−t L(D)

i δy(x) if x, y ∈ Bi , t ≥ 00 otherwise.

Here, δy(x) is the 1-normalized delta function as before. We call pi the restricted heatkernels and note that each pi satisfies(

L + ddt

)pi

t (x, y) = 0 for all x, y ∈ Bi , t ≥ 0.

Furthermore, by Proposition 2.6 in [37]

limi→∞ pi

t (x, y) = pt (x, y) for all x, y ∈ V .

Therefore, to prove a property of the heat kernel it often suffices to prove the correspondingproperty for the reduced heat kernels and then pass to the limit. This is used repeatedly below.

In order to prove Theorem 2, we need a version of the minimum principle for the heatequation in our setting. For U ⊆ V we let U c = V \ U .

Lemma 3.5 (Minimum principle for the heat equation) Let (V, b, c, m) be a weighted graph,U ⊂ V a connected proper subset and u : V × [0, T ] → R such that u(x, ·) is continuouslydifferentiable for every x ∈ U. Suppose that

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(a) (L + ddt )u ≥ 0 on U × [0, T ],

(b) the negative part u− := minu, 0 of u attains its minimum on U × [0, T ],(c) u ≥ 0 on (U c × [0, T ]) ∪ (U × 0).Then, u ≥ 0 on U × [0, T ].Proof Let (x, t) be the minimum of u− on U × [0, T ]. If u(x, t) ≥ 0, then we are done.Furthermore, by (c), we may assume that t > 0. Therefore, suppose that u(x, t) < 0 wheret > 0. As (x, t) is a minimum for u, we have that ( d

dt u)(x, t) = 0 if t ∈ (0, T ) and( d

dt u)(x, t) ≤ 0 if t = T . Since (x, t) is also a minimum with respect to x it follows that(L + d

dt

)u(x, t) ≤ Lu(x, t) = 1

m(x)

∑y∈V

b(x, y) (u(x, t) − u(y, t)) + c(x)m(x)

u(x, t) ≤ 0.

Therefore, by (a), Lu(x, t) = 0 so that u(x, t) = u(y, t) < 0 for all y ∼ x . Repeating theargument we eventually reach some y ∈ U contradicting (c).

We also need the following extension of Lemma 3.10 from [56] which states that the heatkernel on a weakly spherically symmetric graph decays with respect to r . The proof can becarried over directly to our situation so we omit it here.

Lemma 3.6 (Heat kernel decay [56, Lemma 3.10]) Let (V, b, m) be a weighted graph, p bethe heat kernel and pi be the restricted kernels of Bi . Assume that pi

t (x0, ·) is a sphericallysymmetric function for all t ≥ 0 with respect to some vertex x0. Then, given 0 ≤ r ≤ i , wehave for all t > 0

pit (r) > pi

t (r + 1)

and, in general, for all r ∈ N0 and t ≥ 0

pt (r) ≥ pt (r + 1).

With these preparations, we can now prove the second main theorem as follows:

Proof of Theorem 2 Let psym,i be the restricted heat kernels of the weakly spherically sym-metric graph (V sym, bsym, msym) given in the statement of the theorem. On the generalweighted graph (V, b, m), we define the functions ρi

t : V → R, t ≥ 0, by

ρit (x) := psym,i

t (r) for x ∈ Sr (x0), 0 ≤ r ≤ i

and ρit (x) := 0 otherwise.

Under the assumptions of stronger curvature growth and using the heat kernel decay,Lemma 3.6, it follows from the action of L on spherically symmetric functions given afterDefinition 2.3 that, for all 0 ≤ r ≤ i and x ∈ Sr (x0) ⊂ V ,

Lρit (x) = κ+(x)(psym,i

t (r) − psym,it (r + 1)) + κ−(x)(psym,i

t (r) − psym,it (r − 1))

≥ κsym+ (r)(psym,i

t (r) − psym,it (r + 1)) + κ

sym− (r)(psym,i

t (r) − psym,it (r − 1))

= Lsym psym,it (r).

Hence, (L + ddt )ρ

it (x) ≥ (Lsym + d

dt )psym,it (r) = 0.

Let pi be the restricted heat kernels of the general weighted graph (V, b, m) so that(L + d

dt )pit (x0, ·) = 0 on Bi ⊂ V and let u(·, t) = ρi

t (·) − pit (x0, ·). It follows that(

L + ddt

)u(x, t) ≥ 0

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on Bi ×[0, T ]. By compactness, the negative part of u(x, t) attains its minimum on Bi ×[0, T ].Furthermore, as ρi

t = 0 on Bci by definition, we have u(x, t) = 0 on Bc

i and ρi0(·) = pi

0(x0, ·)which is 1

m(x0)= 1

msym(o)for x0 and 0 otherwise. Hence, by the minimum principle, Lemma

3.5, we get that

psym,it (r) = ρi

t (x) ≥ pit (x0, x) for x ∈ Bi .

The desired result now follows by letting i → ∞. The inequality in the case of weakercurvature growth is proven analogously.

4 Spectral estimates

In this section we give an estimate for the bottom of the spectrum and a criterion for discrete-ness of the spectrum in the weakly spherically symmetric case. We then use the heat kernelcomparisons obtained above to give estimates on the bottom of the spectrum for generalweighted graphs.


Let a weighted graph (V, b, m) be given. Let σ(L) denote the spectrum of L and

λ0 := λ0(L) := inf σ(L).

We call λ0 the bottom of the spectrum or the ground state energy. The ground state energycan be obtained by the Rayleigh-Ritz quotient (see, for instance, [47]) as follows:

λ0 = inff ∈D(Q)

Q( f, f )

〈 f, f 〉 = inff ∈Cc(V )

〈L f, f 〉〈 f, f 〉

where the last equality follows since Cc(V ) is dense in D(Q) with respect to 〈·, ·〉Q =〈·, ·〉 + Q(·, ·) as discussed in Sect. 2. Furthermore, the spectrum of an operator may bedecomposed as a disjoint union as follows:

σ(L) = σdisc(L) ∪ σess(L)

where σdisc(L) denotes the discrete spectrum of L , defined as the set of isolated eigenvaluesof finite multiplicity, and σess(L) denotes the essential spectrum, given by σess(L) = σ(L) \σdisc(L). We will use the notation λess

0 (L) to denote the bottom of the essential spectrum ofL .

Fix a vertex x0 and let Sr = Sr (x0) and Br = Br (x0).

Definition 4.1 For f : V → R and r ∈ N0, we define the weighted volume of a ball by

V f (r) =∑x∈Br

f (x)m(x).

In particular, V1(r) = m(Br ). Moreover, for r ∈ N0, we let

∂ B(r) =∑x∈Sr

κ+(x)m(x)

be the measure of the boundary of a ball which is the weight of the edges leaving the ball.Note that, in the weakly spherically symmetric case, ∂ B(r) = κ

sym+ (r)msym(Sr ).

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The first main result of this section is an estimate on the bottom of the spectrum and acriterion for discreteness of the spectrum in terms of volume and boundary growth for weaklyspherically symmetric graphs.

Theorem 3 (Volume and spectrum) Let (V sym, bsym, msym) be a weakly spherically sym-metric graph. If

∞∑r=0

V1(r)

∂ B(r)= a < ∞,

then

λ0(Lsym) ≥ 1

aand σ(Lsym) = σdisc(Lsym).

The second main result of this section is a comparison of the bottom of the spectrum of ageneral weighted graph and a weakly spherically symmetric one.

Theorem 4 (Spectral comparison) If a weighted graph (V, b, m) has stronger (respectively,weaker) curvature growth than a weakly spherically symmetric graph (V sym, bsym, msym),then

λ0(L) ≥ λ0(Lsym) (respectively, λ0(L) ≤ λ0(Lsym)).

If (V sym, bsym, msym) satisfies∑∞

r=0V1(r)∂ B(r)

= a < ∞ and (V, b, m) has stronger curvaturegrowth, then

λ0(L) ≥ 1

aand σ(L) = σdisc(L).

Remarks Let us discuss these results in light of the present literature:

(a) In Sect. 6, we give examples of standard graphs with m ≡ 1 of polynomial volumegrowth satisfying the summability criterion above. For such graphs, it follows that

has positive bottom of the spectrum as well as discrete spectrum. This stands in clearcontrast to the celebrated theorem of Brooks for Riemannian manifolds [9] and resultsfor [22,32] since, for these, subexponential volume growth always implies that thebottom of the essential spectrum is zero.

(b) For statements analogous to Theorem 3 for the Laplacian on spherically symmetricRiemannian manifolds see [2,29].

(c) There are many examples of estimates for the bottom of the spectrum for the graphLaplacian and normalized graph Laplacian , see, for example, [7,15–19,40,44,52,53]. In particular, from the analogue of the Cheeger inequality found in [15] it followsthat λ0() = 0 if ∂ B(r)

V1(r)→ 0 as r → ∞. Our Theorem 3 complements these results

by giving a lower bound for λ0() in the case of unbounded vertex degree. For , ourresult is not applicable as m(x) = deg(x) implies that V1(r) ≥ ∂ B(r).

(d) Discreteness of the spectrum of the graph Laplacians was studied in [23,36,52,56]. Inour context, a characterization for weighted graphs with positive Cheeger’s constant atinfinity to have discrete spectrum was recently given in [38]. In Sect. 6, we discuss howour results complement these and are not implied by any of them, see Corollaries 6.6and 6.7.

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(e) With regards to Theorem 4, it was shown in [10] that the bottom of the spectrum of thegraph Laplacian on a k-regular graph is smaller than that on a k-regular tree. This wasgeneralized from k-regular to arbitrary graphs with degree bounded by k in [52], whichalso contains a corresponding lower bound. Analogous statements for were proven bydifferent means in [58].


The proofs of the first statement in Theorem 3 and the second statement in Theorem 4 arebased on the following characterization for the bottom of the spectrum, which is sometimesreferred to as a Allegretto–Piepenbrink type of theorem. We refer to [27, Theorem 3.1] for aproof and further discussion of earlier results of this kind.

Proposition 4.2 (Characterization of the bottom of the spectrum, [27, Theorem 3.1]) Let(V, b, c, m) be a weighted graph. For α ∈ R the following statements are equivalent:

(i) There exists a non-trivial v : V → [0,∞) such that (L + α)v ≥ 0.(ii) There exists v : V → (0,∞) such that (L + α)v = 0.

(iii) −α ≤ λ0(L).

Therefore, to prove a lower bound on the bottom of the spectrum, it is sufficient to demon-strate a positive (super-)solution to the difference equation above. In the weakly sphericallysymmetric case, we will look for spherically symmetric solutions.

We now state and prove the following lemma which generalizes a result of [57] and givesthe existence of solutions for any initial condition. Note that we allow for a non-negativepotential as we will also use this statement in the next section on stochastic completeness.

Lemma 4.3 (Recursion formula for solutions) Let (V sym, bsym, csym, msym) be a weaklyspherically symmetric graph and α ∈ R. A spherically symmetric function v is a solution to(Lsym + α)v(r) = 0 if and only if

v(r + 1) − v(r) = 1

∂ B(r)

r∑j=0

Cq+α( j)v( j)

where Cq+α( j) = (qsym( j) + α)msym(S j ). In particular, v is uniquely determined by thechoice ofv(0). Consequently, ifv(0) > 0 andα > 0, thenv is a strictly positive, monotonouslyincreasing solution.

Proof The proof is by induction. For r = 0, (Lsym + α)v(0) = 0 gives

(Lsym + α)v(0) = κsym+ (0) (v(0) − v(1)) + (

qsym(0) + α)v(0) = 0

which yields the assertion.Assume now that the recursion formula holds for r − 1 where r ≥ 1. Then,

(Lsym + α)v(r) = 0 reads as

κsym+ (r) (v(r) − v(r + 1)) + κ

sym− (r) (v(r) − v(r − 1)) + (

qsym(r) + α)v(r) = 0.

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Therefore,

v(r + 1) − v(r) = κsym− (r)

κsym+ (r)

(v(r) − v(r − 1)) + 1

κsym+ (r)

(qsym(r) + α

)v(r)

= κsym− (r)

κsym+ (r)

⎛⎝ 1

κsym+ (r − 1)msym(Sr−1)

r−1∑j=0

Cq+α( j)v( j)

⎞⎠

+ 1

κsym+ (r)

(qsym(r) + α

)v(r)

= 1

∂ B(r)

r∑j=0

Cq+α( j)v( j)

by κsym+ (r − 1)msym(Sr−1) = κ

sym− (r)msym(Sr ) as noted in (2.1).

Whenever α > 0, the right hand side of the recursion formula is positive from the assump-tion that v(0) > 0 which gives the monotonicity statement.

In order to prove the statements of Theorems 3 and 4 concerning the essential spectrum,we need to restrict our operator L to the complements of balls. For i ∈ N0, let Bc

i = V \ Bi

and mci be the restriction of m to Bc

i . We restrict the form Q to Cc(Bci ) and take the closure in

2(Bci , mc

i ) with respect to 〈·, ·〉Q . By standard theory, we obtain an operator on 2(Bci , mc

i )

which we call the restriction of L with Dirichlet boundary conditions and which we denoteby L(D)

i . Note that, in contrast to the previous section, these operators are now defined onthe complement of balls and hence on infinite dimensional spaces.

Lemma 4.4 (Existence of strictly positive solutions) Let (V sym, bsym, msym) be a weaklyspherically symmetric graph. Suppose that a = ∑∞

r=0V1(r)∂ B(r)

< ∞. Then, there exists astrictly positive, strictly monotone decreasing spherically symmetric solution v on V sym to(Lsym − 1

a )v = 0 which satisfies

v(r + 1) ≥ 1 − 1

a

r∑j=0

V1( j)

∂ B( j)for all r ∈ N0.

Moreover, for all i ∈ N0, there exists a strictly positive, strictly monotone decreasing functionvi on Bc

i solving (L(D)i − 1

ai)vi = κ

sym− (i + 1)1Si+1 which satisfies

vi (r + 1) ≥ 1 − 1

ai

r∑j=i+1

V i1 ( j)

∂ B( j)for all r ≥ i + 1,

where 1Si+1(x) is 1 for x ∈ Si+1 and 0 otherwise, ai = ∑∞j=i+1

V1( j)∂ B( j) and V i

1 ( j) =msym(B j \ Bi ).

Proof For α = − 1a and csym ≡ 0 the recursion formula of Lemma 4.3 reads as

v(r + 1) − v(r) = − 1

a∂ B(r)

r∑j=0

C1( j)v( j),

where C1( j) = msym(S j ). Hence, there exists a solution v of the equation for v(0) > 0. Inorder to prove our assertion, we show by induction that for all r ∈ N0,

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(i) v(r + 1) < v(r)

(ii) v(r + 1) ≥(

1 − 1a

∑rj=0

V1( j)∂ B( j)

)v(0)

(iii) v(r + 1) > 0.

For r = 0, we get from the recursion formula above that

v(1) − v(0) = − 1

a∂ B(0)msym(0)v(0) < 0

which gives (i). Furthermore,

v(1) =(

1 − V1(0)

a∂ B(0)

)v(0)

which gives (ii) and (iii) follows by the choice of a.Now, suppose that (i), (ii), and (iii) hold for r > 0. Then, since v( j) > 0 for j =

0, 1, . . . , r , the recursion formula above yields that v(r + 1) − v(r) < 0 which gives (i).Furthermore,

v(r + 1) = v(r) − 1

a∂ B(r)

r∑j=0

C1( j)v( j) > v(r) − V1(r)

a∂ B(r)v(0)

≥⎛⎝1 − 1

a

r−1∑j=0

V1( j)

∂ B( j)

⎞⎠ v(0) − V1(r)

a∂ B(r)v(0) =

⎛⎝1 − 1

a

r∑j=0

V1( j)

∂ B( j)

⎞⎠ v(0)

which yields (ii) and (iii) follows by the choice of a.For the second statement, we define the function vi on Bc

i by vi (i + 1) = 1 and

vi (r) = vi (r − 1) − 1

ai∂ B(r − 1)

r−1∑j=i+1

C1( j)vi ( j) for r > i + 1.

By a direct calculation one checks that (L(D)i − 1

ai)vi (i + 1) = κ−(i + 1) ≥ 0 and, as

in the proof of Lemma 4.3, that (L(D)i − 1

ai)vi (r) = 0 for r > i + 1. Now, by the same

arguments as above, one shows that vi is strictly monotone decreasing, satisfies vi (r) ≥1 − 1

ai

∑r−1j=i+1

V i1 ( j)

∂ B( j) and is strictly positive. Proof of Theorem 3 By Lemma 4.4, there is a strictly positive solution to (Lsym − 1

a )v = 0

where a = ∑∞r=0

V1(r)∂ B(r)

. This proves that λ0(Lsym) ≥ 1a by the characterization of the bottom

of the spectrum, Proposition 4.2.We now show that σ(Lsym) = σdisc(Lsym). By standard theory (see, for example, Propo-

sition 18 in [38]) if follows that

λess0 (Lsym) = lim

i→∞ λ0(L(D)i ).

By Proposition 4.2 and the second part of Lemma 4.4, we have that λ0(L(D)i ) ≥ 1

ai. Since

ai → 0 as i → ∞, it follows that λ0(L(D)i ) → ∞ as i → ∞ so that σess(Lsym) = ∅, that is,

σ(Lsym) = σdisc(Lsym). In order to prove the spectral comparison, Theorem 4, we need an analogue of the well-

known theorem of Li which links large time heat kernel behavior and the ground state energy[11,41,46]. It was recently proven for our setting in [28,39].

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Proposition 4.5 (Heat kernel convergence to ground state energy [28,39]) Let (V, b, c, m)

be a weighted graph. For all vertices x and y,

limt→∞

ln pt (x, y)

t= −λ0(L).

Theorem 4 now follows directly:

Proof of Theorem 4 The first statement follows by combining Theorem 2 and Proposi-tion 4.5. For the second statement, we first derive λ0(L) ≥ 1

a from the first statementand Theorem 3. Now, let vi be the strictly positive monotone decreasing functions whichsolve (L(D) − 1

ai)vi = κ

sym− (i +1)1Si+1 on Bc

i ⊆ V sym with ai = ∑∞j=i+1

V1( j)∂ B( j) which exist

according to Lemma 4.4 for all i ≥ 0. We choose vi to be normalized by letting vi (i +1) = 1.We define wi on Bc

i ⊆ V via wi (x) = vi (r) for x ∈ Sr , r ≥ i + 1. Clearly, by the stronger

curvature growth, (L(D)i − 1

ai)wi (x) ≥ (Lsym,(D)

i − 1ai

)vi (r) for x ∈ Sr , r > i + 1. On theother hand, for r = i + 1 and x ∈ Sr we have, again by the stronger curvature growth, that

(L(D)i − 1

ai)wi (x) ≥ (Lsym,(D)

i − 1ai

)vi (i + 1) + (κ−(x) − κsym− (i + 1)) = κ−(x) ≥ 0

since vi (i + 1) = 1. Hence, (L(D)i − 1

ai)wi ≥ 0 with wi strictly positive and we conclude

that λ0(L(D)i ) ≥ 1

aiby Proposition 4.2. As ai → 0, we get, as in the proof of Theorem 3,

that σess(L) = ∅ and, therefore, σ(L) = σdisc(L).

5 Stochastic completeness


The study of the uniqueness of bounded solutions for the heat equation has a long historyin both the discrete, see, for example, [20,48], and the continuous, see [25] and referencestherein, settings. In recent years, there has been interest in finding geometric conditions forinfinite graphs implying this uniqueness, see, for example, [16,19,26,33,37,38,54–57]. Inthe general setting of [37] it is shown that this uniqueness is equivalent to several otherproperties as we discuss below.

Let a weighted graph (V, b, c, m) be given. We let u0 : V → R be bounded and callu : V × [0,∞) → R a solution of the heat equation with initial condition u0 if, for allx ∈ V , u(x, ·) is continuous on [0,∞), differentiable on (0,∞) and satisfies (

L + ddt

)u(x, t) = 0 for x ∈ V, t > 0,

u(x, 0) = u0(x) for x ∈ V .

The question of the uniqueness of bounded solutions for the heat equation on (V, b, c, m) isthen reduced to having u ≡ 0 be the only bounded solution for the heat equation with initialcondition u0 ≡ 0.

In order to study this question, the following function which was introduced in [37] turnsout to be essential. Let

Mt (x) = e−t L 1(x) +t∫

0

e−sLq(x)ds,

where q = cm is the normalized potential, 1 denotes the function whose value is 1 on all

vertices and e−t L is the operator semigroup extended to the space of bounded functions on V .

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The function e−sLq is defined as the pointwise limit along the net of functions g ∈ Cc(V )

such that 0 ≤ g ≤ q , where the net is considered with respect to the natural ordering g ≺ hwhenever g ≤ h. As shown in [37], this limit always exists and 0 ≤ Mt (x) ≤ 1 for all x ∈ V .

The function Mt consists of two parts which can be interpreted as follows: the first term,e−t L 1(x), is the heat which is still in the graph at time t . The integral denotes the heat whichwas killed by the potential up to time t . Thus, 1 − Mt can be interpreted as the heat which istransported to the boundary of the graph.

In this setting, Theorem 1 of [37] (see also Proposition 28 of [38]) states the following:

Proposition 5.1 (Characterization of stochastic completeness [37, Theorem 1]) Let (V, b,

c, m) be a weighted graph. The following statements are equivalent:

(i) There exists v : V → [0,∞) non-zero, bounded such that (L + α)v ≤ 0 for some(equivalently, all) α > 0.

(ii) There exists v : V → (0,∞) bounded such that (L + α)v = 0 for some (equivalently,all) α > 0.

(iii) Mt (x) < 1 for some (equivalently, all) x ∈ V and t > 0.

(iv) There exists a non-trivial, bounded solution to the heat equation with initial conditionu0 ≡ 0.

Such weighted graphs are called stochastically incomplete at infinity. Otherwise, aweighted graph is called stochastically complete at infinity. This extends the usual notion ofstochastic completeness for the Laplacian to the case where a potential is present. Clearly,as already discussed in [37], when the potential is zero, a stochastically complete weightedgraph is also stochastically complete at infinity.

In order to formulate our stochastic completeness comparison theorems, we need to com-pare the potentials of two weighted graphs, continuing Definition 3.2.

Definition 5.2 We say that a weighted graph (V, b, c, m) has stronger (respectively,weaker) potential with respect to x0 ∈ V than a weakly spherically symmetric graph(V sym, bsym, csym, msym) if, for all x ∈ Sr (x0) ⊂ V and r ∈ N0,

q(x) ≥ qsym(r)(respectively, q(x) ≤ qsym(r)

).


It is desirable to have conditions which imply stochastic completeness or incompleteness atinfinity and this is our goal. We start with a characterization of stochastic completeness atinfinity for weakly spherically symmetric graphs whose proof will be given at the end of thesection. It generalizes a result for the graph Laplacian on spherically symmetric graphsfound in [57].

Theorem 5 (Geometric characterization of stochastic completeness) A weakly sphericallysymmetric graph (V sym, bsym, csym, msym) is stochastically complete at infinity if and onlyif

∞∑r=0

Vq+1(r)

∂ B(r)= ∞

where

Vq+1(r) =∑x∈Br

(qsym(x) + 1)msym(x) and ∂ B(r) = κsym+ (r)msym(Sr ).

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Combining this with Theorem 3 we get an immediate corollary, which is an analogue totheorems found in [2,29] for the Laplacian on a Riemannian manifold. The proof is givenright after the proof of Theorem 5.

Corollary 5.3 If a weakly spherically symmetric graph (V sym, bsym, csym, msym) is stochas-tically incomplete at infinity, then

λ0(Lsym) > 0 and σ(Lsym) = σdisc(Lsym).

Remarks (a) The converse statements do not hold. For example, in the standard case, both

and on regular trees of degree greater than 2 have positive bottom of the spectrum butgovern stochastically complete processes. Furthermore, as shown in [36,52], for a treeone has σ() = σdisc() whenever the vertex degree goes to infinity along any sequenceof vertices which eventually leaves every finite set, while stochastic incompletenessrequires that the vertex degree goes to infinity at a certain rate as shown in [56] (see alsoSect. 6).

(b) The statements of the corollary do not hold for general weighted graphs. This can be seenfrom stability results for stochastic incompleteness at infinity proven in [33,37,56] whichstate that attaching any graph to a graph which is stochastically incomplete at infinity ata single vertex does not change the stochastic incompleteness. Therefore, starting witha stochastically incomplete spherically symmetric tree, attachment of a single path toinfinity can drive the bottom of the spectrum down to zero and add essential spectrum(as follows by general principles) without effecting the stochastic incompleteness.

(c) The two statements of the corollary are not completely independent: if a Laplacian ona graph has purely discrete spectrum and the constant function 1 does not belong to2(V, m) or c ≡ 0, then the lowest eigenvalue cannot be zero.

Our second main result of this section is a comparison theorem for stochastic completenessin the spirit of [34].

Theorem 6 (Stochastic completeness at infinity comparison) If a weighted graph (V, b,

c, m) has stronger curvature growth and weaker potential (respectively, weaker curvaturegrowth and stronger potential) than a weakly spherically symmetric graph (V sym, bsym, csym,

msym) which is stochastically incomplete (respectively, complete) at infinity, then (V, b, c, m)

is stochastically incomplete (respectively, complete) at infinity.

Remarks (a) Note that a stronger potential can make a stochastically incomplete graph sto-chastically complete at infinity (compare with Theorem 2 in [37]). This is due to thedefinition of Mt . Specifically, the potential kills heat in the graph and, as such, preventsit from being transported to infinity.

(b) For the results above to hold, it suffices that the comparisons hold outside of a finite set.This is due to the fact that stochastic (in) completeness is stable under finite dimensionalperturbations, compare [33,37,56].


We begin with an observation concerning the solutions to the difference equation on weaklyspherically symmetric graphs which we have encountered before.

Lemma 5.4 (Boundedness of spherically symmetric solutions) Let a weakly sphericallysymmetric graph (V sym, bsym, csym, msym) be given. Let α > 0 and v : V sym → (0,∞) be

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a spherically symmetric function such that (Lsym + α)v = 0. Then, v is unbounded if andonly if

∞∑r=0

Vq+1(r)

∂ B(r)= ∞.

Proof We will use the obvious fact that∑∞

r=0Vq+1(r)

∂ B(r)= ∞ if and only if

∑∞r=0

Vq+α(r)

∂ B(r)= ∞

for some (equivalently, all) α > 0.By the recursion formula of Lemma 4.3, the function v satisfies

v(r + 1) − v(r) = 1

∂ B(r)

r∑i=0

Cq+α(i)v(i)

where Cq+α(r) = (qsym(r) + α)msym(Sr ) and consequently is monotonously increasing asv(0) > 0. Hence, it satisfies

v(r + 1) − v(r) ≥ Vq+α(r)

∂ B(r)v(0)

where Vq+α(r) = ∑x∈Br

(qsym(x) + α)msym(x) = ∑ri=0 Cq+α(i). Therefore, if

∑∞r=0

Vq+1(r)

∂ B(r)= ∞, then v(r) = ∑r−1

i=0 (v(i + 1) − v(i)) → ∞ as r → ∞. Hence, v is unboundedin this case.

On the other hand, the recursion formula and monotonicity imply that

v(r + 1) ≤(

1 + Vq+α(r)

∂ B(r)

)v(r) ≤

r∏i=0

(1 + Vq+α(i)

∂ B(i)

)v(0).

Therefore, if∑∞

r=0Vq+1(r)

∂ B(r)< ∞, then

∏∞r=0

(1 + Vq+α(r)

∂ B(r)

)< ∞ so that v is bounded.

We combine the lemma above with the characterizations of Proposition 5.1 in order toprove stochastic incompleteness at infinity of weakly spherically symmetric graphs. For theother direction, we will need the following criterion for stochastic completeness at infinitywhich is an analogue for a criterion of Has′minskiı from the continuous setting [30]. Recently,Huang [33] has proven a slightly stronger version in the case c ≡ 0. For v : V → R, wewrite

v(x) → ∞ as x → ∞whenever for every C ≥ 0 there is a finite set K such that v|V \K ≥ C .

Proposition 5.5 (Condition for stochastic completeness) If on a weighted graph (V, b, c, m)

there exists v such that (L + α)v ≥ 0 and v(x) → ∞ as x → ∞, then (V, b, c, m) isstochastically complete at infinity.

Proof Suppose there is a function 0 ≤ w ≤ 1 that solves (L + α)w = 0. For given C > 0let K ⊂ V be a finite set such that v|V \K ≥ C . Then u = v − Cw satisfies (L + α)u ≥ 0on V and u ≥ 0 on V \ K . As K is finite, the negative part of u attains its minimum on K .Therefore, by a minimum principle, see [37, Theorem 8], u ≥ 0 on K . Hence, v ≥ Cw forall C > 0. This implies that w ≡ 0. By Proposition 5.1, stochastic completeness at infinityfollows.

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Proof of Theorem 5 By Lemma 4.3, there is always a strictly positive, spherically symmetricsolution to (Lsym + α)v = 0 for α > 0. By Lemma 5.4, this solution is bounded if andonly if the sum in the statement of Theorem 5 converges. In the case of convergence, weconclude stochastic incompleteness at infinity by Proposition 5.1. On the other hand, if thesum diverges, the solution is unbounded and satisfies the assumptions of Proposition 5.5 (bythe spherical symmetry). Hence, the graph is stochastically complete at infinity. Proof of Corollary 5.3 If the graph (V sym, bsym, csym, msym) is stochastically incomplete atinfinity, then the sum of Theorem 5 converges. As a consequence, the sum of Theorem 3converges which implies positive bottom of the spectrum and discreteness of the spectrumof the graph (V sym, bsym, msym). Now, a non-negative potential only lifts the bottom of thespectrum (as can be seen from the Rayleigh-Ritz quotient) which gives the statement. Proof of Theorem 6 Let (V sym, bsym, csym, msym) be a weakly spherically symmetric graph.Let v be the spherically symmetric solution with v(0) = 1 given by Lemma 4.3 for α > 0.Then, v is strictly positive and monotonously increasing. Define w : V → (0,∞) on theweighted graph (V, b, c, m) by w(x) = v(r) for x ∈ Sr (x0) ⊂ V .

First, assume the stochastic incompleteness at infinity of (V sym, bsym, csym, msym). Then,by Lemma 5.4 combined with Theorem 5, the function v, and thus w, is bounded. Underthe assumptions that (V, b, c, m) has stronger curvature growth and weaker potential than(V sym, bsym, csym, msym) and, as v is monotonously increasing by Lemma 4.3, the functionw satisfies for x ∈ Sr , r ∈ N,(

L + α)w(x) = κ+(x) (v(r) − v(r + 1)) + κ−(x) (v(r) − v(r − 1)) + (q(x) + α) v(r)

≤ (Lsym + α

)v(r) = 0.

Hence, the graph (V, b, c, m) is stochastically incomplete at infinity, by Proposition 5.1.Assume now that (V sym, bsym, csym, msym) is stochastically complete at infinity. Then,

again by Proposition 5.1, the function v and, thus w, is unbounded. As above, under theassumptions of weaker curvature growth and stronger potential, one checks that(

L + α)w(x) ≥ (

Lsym + α)v(r) = 0

for all x ∈ Sr ⊂ V , r ∈ N0. Since w is unbounded, w(x) → ∞ as x → ∞, therefore,the weighted graph (V, b, c, m) is stochastically complete at infinity by the condition forstochastic completeness, Proposition 5.5.

6 Applications to graph Laplacians

In this section we discuss the results of this paper for standard graphs. Thus, we consider thesituation b : V × V → 0, 1 and c ≡ 0. Chosing m ≡ 1 and m ≡ deg we obtain the twograph Laplacians, that is, on 2(V ) = 2(V, 1) and on 2(V, deg).

Note that, in the setting for , the curvatures κ± : V → N denote the number of edgesconnecting a vertex x , which lies in the sphere of radius r = d(x, x0), to vertices in thespheres of radius r ± 1. Therefore, we can think of κ− and κ+ as the inner and outer vertexdegrees, respectively. On the other hand, for , the curvatures κ± : V → Q are inner andouter vertex degree divided by the degree.

In this situation, we have the following corollaries of Theorem 1 for the heat kernels.

Corollary 6.1 The operator has a spherically symmetric heat kernel if and only if theinner and outer vertex degrees are spherically symmetric.

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Fig. 2 A tree and an antitree

Corollary 6.2 The operator has a spherically symmetric heat kernel if and only if theratio of inner and outer vertex degrees and the vertex degree are spherically symmetric.

Note that, in order to have a spherically symmetric heat kernel for , even less symmetrythan the graphs of Fig. 1 possess is needed.

Next, we fix m ≡ 1 to focus on and give examples of weakly spherically symmetricgraphs which satisfy our summability condition.

We start with the case of spherically symmetric trees. Here, κ−(r) = 1 and, if V (r) :=V1(r), the summability criterion of Theorems 3 and 5 concerns the convergence or divergenceof

∞∑r=0

V (r)

∂ B(r)=

∞∑r=0

|Br ||Sr+1|

which can be seen to be equivalent to the convergence or divergence of∑∞

r=01

κ+(r), see [57].

Using this, it follows easily, as was already shown in [56], that the threshold for the volumegrowth for stochastic completeness of spherically symmetric trees lies at r ! ∼ er log r . Thisalready stands in contrast to the result of Grigor’yan which puts the threshold for stochasticcompleteness of manifolds at er2

[25]. However, this is still in line with the result of Brookswhich yields that subexponential growth implies absence of a spectral gap for manifolds [9].

We now come to the family of spherically symmetric graphs, here called antitrees, whichyield our surprising examples.

Definition 6.3 A standard weakly spherically symmetric graph with m ≡ 1 is called anantitree if κ+(r) = |Sr+1| for all r ∈ N0.

As opposed to trees, which are connected graphs with as few connections as possiblebetween spheres, antitrees have the maximal number of connections. The contrast is illustratedin Fig. 2. Such graphs have already been used as examples in [17,54,57].

In the case of antitrees, the summability criterion concerns the convergence or divergenceof

∞∑r=0

V (r)

∂ B(r)=

∞∑r=0

|Br ||Sr ||Sr+1| .

For functions f, g : N0 → R we write f ∼ g provided that there exist constants c and Csuch that cg(r) ≤ f (r) ≤ Cg(r) for all r ∈ N0. We can then relate the summability criterionabove to volume growth through the following lemma whose proof is immediate.

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Lemma 6.4 (Volume growth of antitrees) Let β > 0. An antitree with κ+(r) ∼ rβ for r ∈ N0

satisfies V (r) ∼ rβ+1. Furthermore, if β > 2 (respectively, β ≤ 2), then

∞∑r=0

V (r)

∂ B(r)< ∞

(respectively,

∞∑r=0

V (r)

∂ B(r)= ∞

).

Combining this lemma with Theorem 5 immediately gives the following corollary.

Corollary 6.5 (Polynomial growth and stochastic incompleteness) Let β > 0. An antitreewith κ+(r) ∼ rβ satisfies V (r) ∼ rβ+1 and is stochastically complete if and only if β ≤ 2.

This phenomenon was already observed in [57] and it gives an even sharper contrast toGrigor’yan’s threshold for manifolds as the volume growth does not even have to be expo-nential for the antitree to become stochastically incomplete. Furthermore, in [26, Theorem1.4], it is shown that any graph whose volume growth is less than cubic is stochasticallycomplete. Thus, our examples are, in some sense, the stochastically incomplete graphs withthe smallest volume growth.

It should also be mentioned that it was recently shown by Huang [33], using ideas foundin [1], that the condition

∑∞r=0

V (r)∂ B(r)

= ∞ which implies stochastic completeness of weaklyspherically symmetric graphs does not imply stochastic completeness for general graphs.

Let us now turn to a discussion of the spectral consequences. Combining the volumegrowth of antitrees, Lemma 6.4, with Theorem 3 gives the following immediate corollary.

Corollary 6.6 (Polynomial growth and positive bottom of the spectrum) Let β > 2. Anantitree with κ+(r) ∼ rβ satisfies V (r) ∼ rβ+1 with

λ0() > 0 and σ() = σdisc().

This corollary shows that, for , there are no direct analogues to Brook’s theorem whichstates, in particular, that subexponential volume growth of manifolds implies that the bottomof the essential spectrum is zero. On the other hand, there is a result by Fujiwara [22] (whichwas later generalized in [32]) for the normalized Laplacian acting on 2(V, deg), whichstates that

λess0 () ≤ 1 − 2e

μ2

1 + eμ.

Here, the exponential volume growth is given by μ = lim supr→∞ 1r log V1(r) where

V1 (r) = ∑x∈Br

deg(x). Therefore, if a graph has subexponential volume growth with respectto the measure deg, it follows that λ0() ≤ λess

0 () = 0.Let us also discuss how our Theorem 3 complements some of the results found in [36,38].

There it is shown that, for graphs with positive Cheeger constant at infinity, α∞ > 0, rapidbranching is equivalent to discreteness of the spectrum of the Laplacian. The constant α∞ isdefined as the limit over the net of finite sets K of the quantities

αK = inf∂W

m(W )

where ∂W is the number of edges leaving W and the infimum is taken over all finite setsW ⊆ V \ K , see [23,36,38]. Now, subexponential volume growth, that is, μ = 0 impliesα∞ = 0 by 1−√

1 − α2∞ ≤ λess0 (), shown in [23], combined with the estimate for λess

0 ()

given above. Corollary 6.6 gives examples of graphs with μ = 0 and thus α∞ = 0 but forwhich has no essential spectrum.

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Another interesting consequence of Theorem 2 and Theorem 4 for is the following:

Corollary 6.7 Let G and G ′ be the graph Laplacians of two weakly spherically sym-metric graphs which have the same curvature growth. Then, λ0(G) = λ0(G ′).

This means, in particular, that the Laplacians on all graphs in Fig. 1 have the samebottom of the spectrum. Note that this is not at all true for the normalized graph Laplacian as it operates on 2(V, m) = 2(V, deg) and the presence of edges connecting vertices onthe same sphere clearly effects the degree measure.

To illustrate this contrast, note that on a k-regular tree Tk the bottom of the spectrumis known to be λ0(Tk ) = 1

k (k − 2√

k − 1) for all k. On the other hand, if one connectsall vertices in each sphere one obtains a graph Gk such that λ0(Gk ) = 0 as shown in [36,Theorem 6]. However, from Corollary 6.7 above, λ0(Tk ) = λ0(Gk ) = k−2

√k − 1 which

is also new compared to [36, Theorem 6], where only λ0(Gk ) ≤ k is shown. In particular,we have another example where the ground state energies of and differ.

Acknowledgments The authors are grateful to Józef Dodziuk for his continued support. MK and RW wouldlike to thank the Group of Mathematical Physics of the University of Lisbon for their generous backing whileparts of this work were completed. In particular, RW extends his gratitude to Pedro Freitas and Jean-ClaudeZambrini for their encouragement and assistance. RW gratefully acknowledges financial support of the FCTin the forms of grant SFRH/BPD/45419/2008 and project PTDC/MAT/101007/2008.

Appendix A: Reducing subspaces and commuting operators

We study symmetries of selfadjoint operators. These symmetries are given in terms ofbounded operators commuting with the selfadjoint operator in question. We present a generalcharacterization in Theorem A.1. With Lemma A.5, we then turn to the question of how sym-metries of a symmetric non-negative operator carry over to its Friedrichs extension. Finally,we specialize to the situation in which the bounded operator is a projection onto a closedsubspace. The main result of this appendix, Corollary A.8, characterizes when a selfadjointoperator commutes with such a projection.

While these results are certainly known in one form or another, we have not found allof them in the literature in the form discussed below. In the main body of the paper theywill be applied to Laplacians on graphs. However, they are general enough to be applied toLaplace-Beltrami operators on manifolds as well.

A subspace U of a Hilbert space is said to be invariant under the bounded operator A ifA maps U into U .

Theorem A.1 Let L be a selfadjoint non-negative operator on the Hilbert space H and A abounded operator on H. Then, the following assertions are equivalent:

(i) D(L) is invariant under A and L Ax = ALx for all x ∈ D(L).(ii) D(L1/2) is invariant under A and L1/2 Ax = AL1/2x for all x ∈ D(L1/2).

(iii) 1[0,t](L)A = A1[0,t](L) for all t ≥ 0.(iv) e−t L A = Ae−t L for all t ≥ 0.(v) (L + α)−1 A = A(L + α)−1 for all α > 0.

(vi) g(L)A = Ag(L) for all bounded measurable g : [0,∞) −→ C.

Proof This is essentially standard. We sketch a proof for the convenience of the reader. Wefirst show that (iii), (iv), (v) and (vi) are all equivalent:

(iii) ⇒ (iv): This follows by a simple approximation argument.

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(iv) ⇒ (v): This follows immediately from (L + α)−1 = ∫ ∞0 e−tαe−t L dt (which, in

turn, is a direct consequence of the spectral calculus).(v) ⇒ (vi): The assumption (v) together with a Stone/Weierstrass-type argument shows

that g(L)A = Ag(L) for all continuous g : [0,∞) −→ C with g(x) → 0 for x → ∞. Now,it is not hard to see that the set

f : [0,∞) −→ C | f measurable and bounded with f (L)A = A f (L)is closed under pointwise convergence of uniformly bounded sequences. This gives thedesired statement (vi).

(vi) ⇒ (iii): This is obvious.We now show (ii) ⇒ (i) ⇒ (v) and (vi) ⇒ (ii).(ii) ⇒ (i): This is clear as L = L1/2 L1/2.(i) ⇒ (v): Obviously, (i) implies A(L + α)x = (L + α)Ax for all α ∈ R and x ∈ D(L).

As (L + α) is injective for α > 0 we infer for all such α that

(L + α)−1 A = A(L + α)−1.

(vi) ⇒ (ii): For every natural number n the operator L1/21[0,n](L) = (id1/21[0,n])(L)

is a bounded operator commuting with A by (vi). Let x ∈ D(L1/2) be given and set xn :=1[0,n](L)x . Then, xn belongs to D(L1/2). Moreover, as 1[0,n](L) is a projection, we obtainby (vi) that

Axn = A1[0,n](L)x = 1[0,n](L)A1[0,n](L)x = 1[0,n](L)Axn .

In particular, Axn belongs to D(L1/2) as well. This gives, by (vi) again, that

L1/2 Axn = L1/21[0,n](L)Axn = AL1/21[0,n](L)x .

As x belongs to D(L1/2), we infer that L1/21[0,n](L)x converges to L1/2x . Moreover, Axn

obviously converges to Ax . As L1/2 is closed, we obtain that Ax belongs to D(L1/2) as welland L1/2 Ax = L1/2 Ax holds. Remark (a) The method to prove (v) ⇒ (vi) can be strengthened as follows: Let L be a

selfadjoint operator with spectrum . Let B() be the algebra of all bounded measurablefunctions on . A sequence ( fn) in B() is said to converge to f ∈ B() in the senseof (♣) if the ( fn) are uniformly bounded and converge pointwise to f . Let F be a subsetof B such that f (L)A = A f (L) holds for all f ∈ F . If the smallest subalgebra ofB which contains F and is closed under convergence with respect to (♣) is B, theng(L)A = Ag(L) for all g ∈ B.

(b) If L is an arbitrary selfadjoint operator then the equivalence of (i), (iii) and (vi) is stilltrue and the semigroup in (iv) can be replaced by the unitary group and the resolvents in(v) can be replaced by resolvents for α ∈ C \ R (as can easily be seen using (a) of thisremark).

Definition A.2 Let L be a selfadjoint non-negative operator on a Hilbert space H and A abounded operator onH. Then, A is said to commute with L if one of the equivalent statementsof the theorem holds.

Corollary A.3 Let L be a selfadjoint non-negative operator on a Hilbert space H and A abounded operator on H. Then, A commutes with L if and only if its adjoint A∗ commuteswith L.

Proof Take adjoints in (iii) of the previous theorem.

123


A simple situation in which the previous theorem can be applied is given next.

Proposition A.4 Let L be a selfadjoint non-negative operator on the Hilbert space H andlet A be a bounded operator on H. Let, for each natural number n, a closed subspace Hn

of H be given with AHn ⊂ Hn and ∪nHn = H. If, for each n, there exists a selfadjointnon-negative operator Ln from Hn to Hn with ALn = Ln A and

(Ln+k + α)−1x → (L + α)−1x, k → ∞,

for all natural numbers n, x ∈ Hn and α > 0, then AL = L A holds. A correspondingstatement holds with resolvents replaced by the semigroup.

Proof By assumption we have A(L + α)−1x = (L + α)−1 Ax for all x from the dense set∪nHn . By boundedness of the respective operators we infer A(L +α)−1 = (L +α)−1 A andthe statement follows from the previous theorem.

The previous theorem deals with symmetries of a selfadjoint operator L . Often, the self-adjoint operator arises as the Friedrichs extension of a symmetric operator. We next studyhow symmetries of a symmetric operator carry over to its Friedrichs extension. Specifically,we consider the following situation:

(*) Let H be a Hilbert space with inner product 〈·, ·〉. Let L0 be a symmetric operator onH with domain D0. Let Q0 be the associated form, i.e., Q0 is defined on D0 × D0

via Q0(u, v) := 〈L0u, v〉. Assume that Q0 is non-negative, i.e., Q0(u, u) ≥ 0 for allu ∈ D0. Then, Q0 is closable. Let Q be the closure of Q0, D(Q) the domain of Q andL the Friedrichs extension of L0, i.e., L is the selfadjoint operator associated to Q.

Lemma A.5 Assume (∗). Let A be a bounded operator on H with D0 invariant under A andA∗, AL0x = L0 Ax and A∗L0x = L0 A∗x for all x ∈ D0. Then, the following assertionsare equivalent:

(i) D(L) is invariant under A and ALx = L Ax for all x ∈ D(L).(ii) D(Q) is invariant under A and A∗ and Q(Ax, y) = Q(x, A∗y) for all x, y ∈ D(Q).

(iii) There exists a C ≥ 0 with both Q0(Ax, Ax) ≤ C Q0(x, x) and Q0(A∗x, A∗x) ≤C Q0(x, x) for all x ∈ D0.

Proof (iii) ⇒ (ii): By AL0x = L0 Ax for all x ∈ D0 we infer that Q0(Ax, y) = Q0(x, A∗y)

for all x, y ∈ D0. As Q is the closure of Q0, it now suffices to show that both (Aun) and(A∗un) are a Cauchy sequences with respect to the Q-norm, whenever (un) is a Cauchysequence with respect to the Q-norm in D0. This follows directly from (iii).

(ii) ⇒ (i): Let x ∈ D(L) be given. Then, x belongs to D(Q) and, by (ii), Ax belongs toD(Q) as well. Thus, we can calculate for all y ∈ D(Q)

Q(Ax, y) = Q(x, A∗y) = 〈Lx, A∗y〉 = 〈ALx, y〉.This implies that Ax ∈ D(L) and L Ax = ALx . Hence, we obtain (i).

(i) ⇒ (iii): From Theorem A.1 and (i) we infer that L1/2 Ax = AL1/2x for all x ∈D(L1/2). Now, for x ∈ D0 it holds that

Q0(x, x) = 〈L0x, x〉 = 〈L1/2x, L1/2x〉 = ‖L1/2x‖2.

By Ax ∈ D0 for x ∈ D0 a direct calculation gives

Q0(Ax, Ax) = ‖L1/2 Ax‖2 = ‖AL1/2x‖2 ≤ ‖A‖2‖L1/2x‖2 = ‖A‖2 Q0(x, x).

A similar argument shows Q0(A∗x, A∗x) ≤ ‖A∗‖2 Q0(x, x). This finishes the proof.

123


We now turn to the special situation that A is the projection onto a closed subspace. Inthis case, some further strengthening of the above result is possible. We first provide anappropriate definition.

Definition A.6 Let H be a Hilbert space and S a symmetric operator on H with domainD(S). A closed subspace U ofH with associated orthogonal projection P is called a reducingsubspace for S if D(S) is invariant under P and S Px = P S Px for all x ∈ D(S).

The previous definition is just a commutation condition in the form discussed above asshown in the next lemma.

Lemma A.7 Let S be a symmetric operator on the Hilbert space H and P be the orthogonalprojection onto a closed subspace U of H. Then, the following assertions are equivalent:

(i) U is a reducing subspace for S.(ii) D(S) is invariant under P and S Px = P Sx holds for all x ∈ D(S).

Proof The implication (ii) ⇒ (i) is obvious. It remains to show (i) ⇒ (ii): We first showP Sy = 0 for all y ∈ D(S) with Py = 0 (i.e., y ⊥ U ): Choose x ∈ D(S) arbitrarily. Then,as Px ∈ D(S) ⊂ D(S∗) we obtain

〈P Sy, x〉 = 〈Sy, Px〉 = 〈y, S∗ Px〉 = 〈y, S Px〉 = 〈y, P S Px〉 = 〈Py, S Px〉 = 0.

As D(S) is dense, we infer P Sy = 0. Let now x ∈ D(S) be arbitrary. Then, x = Px + (1 −P)x and both Px and (1 − P)x belong to D(S). Thus, we can calculate

P Sx = P S Px + P S(1 − P)x = P S Px = S Px .

This finishes the proof. We now come to the main result of the appendix dealing with symmetries of symmetric

operators in terms of reducing subspaces.

Corollary A.8 Assume (∗). Let U be a closed subspace ofH and A the orthogonal projectiononto U. Assume that D0 is invariant under A. Then, the following assertions are equivalent:

(i) U is a reducing subspace for L0, i.e., L0 Ax = AL0x for all x ∈ D0.(ii) Q0(Ax, Ay) = Q0(Ax, y) = Q0(x, Ay) for all x, y ∈ D0.

(iii) D(Q) is invariant under A and Q(Ax, Ay) = Q(Ax, y) = Q(x, Ay) for all x, y ∈D(Q).

(iv) U is a reducing subspace for L.(v) A commutes with e−t L for every t ≥ 0.

(vi) A commutes with (L + α)−1 for any α > 0.

Proof Obviously, (i) and (ii) are equivalent. The equivalence of (iii) and (iv) follows from theequivalence of (i) and (ii) in Lemma A.5. The equivalence between (iv), (v) and (vi) followsimmediately from Theorem A.1. The implication (iii) ⇒ (ii) is clear (as AD0 ⊆ D0). Itremains to show (ii) ⇒ (iii): A direct calculation using (ii) gives for all x ∈ D0 that

Q0(x, x) = Q0((A + (1 − A))x, x)

= Q0(Ax, x) + Q0((1 − A)x, x)

= Q0(Ax, Ax) + Q0((1 − A)x, (1 − A)x).

123


This shows

Q0(Ax, Ax) ≤ Q0(x, x)

for all x ∈ D0. Now, the implication (iii) ⇒ (ii) from Lemma A.5 gives (iii).

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CHAPTER 7

M. Bonnefont, S. Golenia, M. Keller, Eigenvalueasymptotics for Schrodinger operators on sparse

graphs, arXiv:1311.7221.

169

EIGENVALUE ASYMPTOTICS FOR SCHRÖDINGEROPERATORS ON SPARSE GRAPHS

MICHEL BONNEFONT, SYLVAIN GOLÉNIA, AND MATTHIAS KELLER

Abstract. We consider Schrödinger operators on sparse graphs. The geomet-ric definition of sparseness turn out to be equivalent to a functional inequalityfor the Laplacian. In consequence, sparseness has in turn strong spectral andfunctional analytic consequences. Specifically, one consequence is that it allowsto completely describe the form domain. Moreover, as another consequence itleads to a characterization for discreteness of the spectrum. In this case wedetermine the first order of the corresponding eigenvalue asymptotics.

1. Introduction

The spectral theory of discrete Laplacians on finite or infinite graphs has drawna lot of attention for decades. One important aspect is to understand the relationsbetween the geometry of the graph and the spectrum of the Laplacian. Often aparticular focus lies on the study of the bottom of the spectrum and the eigenvaluesbelow the essential spectrum.

Certainly the most well-known estimates for the bottom of the spectrum ofLaplacians on infinite graphs are so called isoperimetric estimates or Cheeger in-equalities. Starting with [D1] in the case of infinite graphs, these inequalities wereintensively studied and resulted in huge body of literature, where we here mentiononly [BHJ, BKW, D2, DK, F, M1, M2, K1, KL2, Woj1]. In certain more spe-cific geometric situations the bottom of the spectrum might be estimated in termsof curvature, see [BJL, H, JL, K1, K2, KP, LY, Woe]. There are various othermore recent approaches such as Hardy inequalities in [G] and summability criteriainvolving the boundary and volume of balls in [KLW].

In this work we focus on sparse graphs to study discreteness of spectrum andeigenvalue asymptotics. In a moral sense, the term sparse means that there arenot ‘too many’ edges, however, throughout the years various different definitionswere investigated. We mention here [EGS, L] as seminal works which are closelyrelated to our definitions. As it is impossible to give a complete discussion ofthe development, we refer to some selected more recent works such as [AABL, B,LS, M2] and references therein which also illustrates the great variety of possibledefinitions. Here, we discuss three notions of sparseness that result in a hierarchyof very general classes of graphs.

Let us highlight the work of Mohar [M3], where large eigenvalues of the adja-cency matrix on finite graphs are studied. Although our situation of infinite graphs

Date: January 24, 2014.2000 Mathematics Subject Classification. 47A10, 34L20,05C63, 47B25, 47A63.Key words and phrases. discrete Laplacian, locally finite graphs, eigenvalues, asymptotic, pla-

narity, sparse, functional inequality.1

2 M. BONNEFONT, S. GOLÉNIA, AND M. KELLER

with unbounded geometry requires fundamentally different techniques – functionalanalytic rather than combinatorial – in spirit our work is certainly closely related.

The techniques used in this paper owe on the one hand to considerations ofisoperimetric estimates as well as a scheme developed in [G] for the special case oftrees. In particular, we show that a notion of sparseness is a geometric characteri-zation for an inequality of the type

(1− a) deg−k ≤ ∆ ≤ (1 + a) deg +k

for some a ∈ (0, 1), k ≥ 0 which holds in the form sense (precise definitions anddetails will be given below). The moral of this inequality is that the asymptoticbehavior of the Laplacian ∆ is controlled by the vertex degree function deg (thesmaller a the better the control).

Furthermore, such an inequality has very strong consequences which follow fromwell-known functional analytic principles. These consequences include an explicitdescription of the form domain, characterization for discreteness of spectrum andeigenvalue asymptotics.

Let us set up the framework. Here, a graph G is a pair (V ,E ), where V denotesa countable set of vertices and E : V × V → 0, 1 is a symmetric function withzero diagonal determining the edges. We say two vertices x and y are adjacent orneighbors whenever E (x, y) = E (y, x) = 1. In this case, we write x ∼ y and wecall (x, y) and (x, y) the (directed) edges connecting x and y. We assume that G islocally finite that is each vertex has only finitely many neighbors. For any finite setW ⊆ V , the induced subgraph GW := (W ,EW ) is defined by setting EW := E |W ×W ,i.e., an edge is contained in GW if and only if both of its vertices are in W .

We consider the complex Hilbert space `2(V ) := ϕ : V → C such that∑x∈V |ϕ(x)|2 < ∞ endowed with the scalar product 〈ϕ,ψ〉 :=

∑x∈V ϕ(x)ψ(x),

ϕ,ψ ∈ `2(V ).For a function g : V → C, we denote the operator of multiplication by g on

`2(V ) given by ϕ 7→ gϕ and domain D(g) := ϕ ∈ `2(V ) | gϕ ∈ `2(V ) with slightabuse of notation also by g.

Let q : V → [0,∞). We consider the Schrödinger operator ∆ + q defined as

D(∆ + q) :=ϕ ∈ `2(V ) |

(v 7→

∑

w∼v(ϕ(v)− ϕ(w)) + q(v)ϕ(v)

)∈ `2(V )

(∆ + q)ϕ(v) :=∑

w∼v(ϕ(v)− ϕ(w)) + q(v)ϕ(v).

The operator is non-negative and selfadjoint as it is essentially selfadjoint on Cc(V ),the set of finitely supported functions V → R, (confer [Woj1, Theorem 1.3.1], [KL1,Theorem 6]). In Section 2 we will allow for potentials whose negative part is formbounded with bound strictly less than one. Moreover, in Section 4 we consider alsomagnetic Schrödinger operators.

As mentioned above sparse graphs have already been introduced in various con-texts with varying definitions. In this article we also treat various natural general-izations of the concept. In this introduction we stick to an intermediate situation.

Definition. A graph G := (V ,E ) is called k-sparse if for any finite set W ⊆ Vthe induced subgraph GW := (W ,EW ) satisfies

2|EW | ≤ k|W |,

EIGENVALUE ASYMPTOTICS FOR SCHRÖDINGER OPERATORS ON SPARSE GRAPHS 3

where |A| denotes the cardinality of a finite set A and we set

|EW | :=1

2|(x, y) ∈ W ×W | EW (x, y) = 1|,

that is we count the non-oriented edges in GW .

Examples of sparse graphs are planar graphs and, in particular, trees. We referto Section 6 for more examples.

For a function g : V → R and a finite set W ⊆ V , we denote

g(W ) :=∑

x∈W

g(x).

Moreover, we define

lim inf|x|→∞

g(x) := supW ⊂V finite

infx∈V \W

g(x), lim sup|x|→∞

g(x) := infW ⊂V finite

supx∈V \W

g(x).

For two selfadjoint operators T1, T2 on a Hilbert space and a subspace D0 ⊆ D(T1)∩D(T2) we write T1 ≤ T2 on D0 if 〈T1ϕ,ϕ〉 ≤ 〈T2ϕ,ϕ〉 for all ϕ ∈ D0. Moreover, for aselfadjoint semi-bounded operator T on a Hilbert space, we denote the eigenvaluesbelow the essential spectrum by λn(T ), n ≥ 0, with increasing order counted withmultiplicity.

The next theorem is a special case of the more general Theorem 2.2 in Section 2.It illustrates our results in the case of sparse graphs introduced above and includesthe case of trees, [G, Theorem 1.1], as a special case. While the proof in [G] uses aHardy inequality, we rely on some new ideas which have their roots in isoperimetrictechniques. The proof is given in Section 2.2.

Theorem 1.1. Let G := (V ,E ) be a k-sparse graph and q : V → [0,∞). Then,we have the following:

(a) For all 0 < ε ≤ 1,

(1− ε)(deg +q)− k

2

(1

ε− ε)≤ ∆ + q ≤ (1 + ε)(deg +q) +

k

2

(1

ε− ε),

on Cc(V ).(b) D

((∆ + q)1/2

)= D

((deg + q)1/2

).

(c) The operator ∆ + q has purely discrete spectrum if and only if

lim inf|x|→∞

(deg + q)(x) =∞.

In this case, we obtain

lim infλ→∞

λn(∆ + q)

λn(deg + q)= 1.

As a corollary, we obtain following estimate for the bottom and the top of the(essential) spectrum.

Corollary 1.2. Let G := (V ,E ) be a k-sparse graph and q : V → [0,∞). Defined := infx∈V (deg +q)(x) and D := supx∈V (deg +q)(x). Assume d < k ≤ D < +∞,then

d− 2

√k

2

(d− k

2

)≤ inf σ(∆ + q) ≤ supσ(∆ + q) ≤ D − 2

√k

2

(D − k

2

).


Define dess := lim inf |x|→∞(deg +q)(x) and Dess := lim sup|x|→∞(deg +q)(x). As-sume dess < k ≤ Dess < +∞, then

dess−2

√k

2

(dess −

k

2

)≤ inf σess(∆+q) ≤ supσess(∆+q) ≤ Dess−2

√k

2

(Dess −

k

2

).

Proof of Corollary 1.2. The conclusion follows by taking ε = min(√

k2d−k , 1

)in

(a) in of Theorem 1.1.

Remark 1.3. The bounds in Corollary 1.2 are optimal for the bottom and the topof the (essential) spectrum in the case of regular trees.

The paper is structured as follows. In the next section an extension of the notionof sparseness is introduced which is shown to be equivalent to a functional inequalityand equality of the form domains of ∆ and deg. In Section 3 we consider almostsparse graphs for which we obtain precise eigenvalue asymptotics. Furthermore,in Section 4 we shortly discuss magnetic Schrödinger operators. Our notion ofsparseness has very explicit but non-trivial connections to isoperimetric inequalitieswhich are made precise in Section 5. Finally, in Section 6 we discuss some examples.

2. A geometric characterization of the form domain

In this section we characterize equality of the form domains of ∆ + q and deg +qby a geometric property. This geometric property is a generalization of the notionof sparseness from the introduction. Before we come to this definition, we introducethe class of potentials that is treated in this paper.

Let α > 0. We say a potential q : V → R is in the class Kα if there is Cα ≥ 0such that

q− ≤ α(∆ + q+) + Cα,

where q± := max(±q, 0). For α ∈ (0, 1), we define the operator ∆ + q via the formsum of the operators ∆ + q+ and −q− (i.e., by the KLMN Theorem, see e.g., [RS,Theorem X.17]). Note that ∆ + q is bounded from below and

D(|∆ + q| 12 ) = D((∆ + q+)12 ) = D(∆

12 ) ∩ D(q

12+),

where |∆ + q| is defined by the spectral theorem. The last equality follows from[GKS, Theorem 5.6]. in the sense of functions and forms.

An other important class are the potentials

K0+ :=⋂

α∈(0,1)

Kα.

In our context of sparseness, we can characterize the class K0+ to be the po-tentials whose negative part q− is morally o(deg +q+), see Corollary 2.9. Let usmention that if q− is in the Kato class with respect to ∆ + q+, i.e., if we havelim supt→0+ ‖e−t(∆+q+)q−‖∞ = 0, then q := q+ − q− ∈ K0+ by [SV, Theorem 3.1].

Next, we come to an extension of the notion of sparseness. For a set W ⊆ V ,let the boundary ∂W of W be the set of edges emanating from W

∂W := (x, y) ∈ W × V \W | x ∼ y.


Definition. Let G := (V ,E ) be a graph and q : V → R. For given a ≥ 0 andk ≥ 0, we say that (G , q) is (a, k)-sparse if for any finite set W ⊆ V the inducedsubgraph GW := (W ,EW ) satisfies

2|EW | ≤ k|W |+ a(|∂W |+ q+(W )).

Remark 2.1. (a) Observe that the definition depends only on q+. The negativepart of q will be taken in account through the hypothesis Kα or K0+ in our theo-rems.(b) If (G , q) is (a, k)-sparse, then (G , q′) is (a, k)-sparse for every q′ ≥ q.(c) As mentioned above there is a great variety of definitions which were so farpredominantly established for (families of) finite graphs. For example it is askedthat |E | = C|V | in [EGS], |EW | ≤ k|W |+ l in [L, LS], |E | ∈ O(|V |) in [AABL] anddeg(W ) ≤ k|W | in [M3].

We now characterize the equality of the form domains in geometric terms.

Theorem 2.2. Let G := (V ,E ) be a graph and q ∈ Kα, α ∈ (0, 1). The followingassertions are equivalent:

(i) There are a, k ≥ 0 such that (G , q) is (a, k)-sparse.(ii) There are a ∈ (0, 1) and k ≥ 0 such that on Cc(V )

(1− a)(deg +q)− k ≤ ∆ + q ≤ (1 + a)(deg +q) + k.

(iii) There are a ∈ (0, 1) and k ≥ 0 such that on Cc(V )

(1− a)(deg +q)− k ≤ ∆ + q.

(iv) D(|∆ + q|1/2) = D(|deg +q|1/2).Furthermore, ∆ + q has purely discrete spectrum if and only if

lim inf|x|→∞

(deg + q)(x) =∞.

In this case, we obtain

1− a ≤ lim infn→∞

λn(∆ + q)

λn(deg +q)≤ lim sup

n→∞

λn(∆ + q)

λn(deg +q)≤ 1 + a.

Before we come to the proof of Theorem 2.2, we summarize the relation be-tween the sparseness parameters (a, k) and the constants (a, k) in the inequality inTheorem 2.2 (ii).

Remark 2.3. Roughly speaking a tends to ∞ as a tends to 1− and a tends to0+ as a tends to 0+ and vice-versa. More precisely, Lemma 2.5 we obtain that forgiven a and k the values of a and k can be chosen to be

a =a

1− a and k =k

1− a .

Reciprocally, given a, k ≥ 0 and q : V → [0,∞), Lemma 2.7 distinguishes the casewhere the graph is sparse a = 0 and a > 0. For a = 0 we may choose a ∈ (0, 1)arbitrary and

k =k

2

(1

a− a).


For an (a, k)-sparse graph with a > 0 the precise constants are found below inLemma 2.7. Here, we discuss the asymptotics. For a→ 0+, we obtain

a '√

2a and k ' k

2a,

and for a→∞a ' 1− 3

8a2and k ' 3k

4a.

In the case q ∈ Kα, the constants a, k from the case q ≥ 0 have to be replaced byconstants whose formula can be explicitly read from Lemma A.3. For α→ 0+, theconstant replacing a tends to a while the asymptotics of the constant replacing kdepend also on the behavior of Cα from the assumption q− ≤ α(∆ + q) + Cα.

Remark 2.4. (a) Observe that in the context of Theorem 2.2 statement (iv) isequivalent to

(iv’) D(|∆ + q|1/2) = D((deg +q+)1/2).

Indeed, (ii) implies the corresponding inequality for q = q+. Thus, as q ∈ Kα,

D(|∆ + q| 12 ) = D((∆ + q+)12 ) = D((deg +q+)

12 ).

(b) The definition of the class K0+ is rather abstract. Indeed, Theorem 2.2 yieldsa very concrete characterization of these potentials, see Corollary 2.9 below.(c) Theorem 2.2 characterizes equality of the form domains. Another natural ques-tion is under which circumstances the operator domains agree. For a discussion onthis matter we refer to [G, Section 4.1].

The rest of this section is devoted to the proof of the results which are dividedinto three parts. The following three lemmas essentially show the equivalences(i)⇔(ii)⇔(iii) providing the explicit dependence of (a, k) on (a, k) and vice versa.The third part uses general functional analytic principles collected in the appendix.

The first lemma shows (iii)⇒(i).

Lemma 2.5. Let G := (V ,E ) be a graph and q : V → R. If there are a ∈ (0, 1)

and k ≥ 0 such that for all f in Cc(V ),

(1− a)〈f, (deg +q)f〉 − k‖f‖2 ≤ 〈f,∆f + qf〉,then (G , q) is (a, k)-sparse with

a =a

1− a and k =k

1− a .

Remark 2.6. We stress that we suppose solely that q : V → R and work with∆|Cc(V ) + q|Cc(V ). We do not specify any self-adjoint extension of the latter.

Proof. Let f ∈ Cc(V ). By adding q− to the assumed inequality we obtain immedi-ately

(1− a)〈f, (deg +q+)f〉 − k‖f‖2 ≤ 〈f,∆f + q+f〉.Let W ⊆ V be a finite set and denote by 1W the characteristic function of the setW . We recall the basic equalities

deg(W ) = 2|EW |+ |∂W | and 〈1W ,∆1W 〉 = |∂W |.


Therefore, applying the asserted inequality with f = 1W , we obtain

2|EW | ≤k

1− a |W |+a

1− a (|∂W |+ q+(W )) .

This proves the statement. The second lemma gives (i)⇒(ii) for q ≥ 0.

Lemma 2.7. Let G := (V ,E ) be a graph and q : V → [0,∞). If there are a, k ≥ 0such that (G , q) is (a, k)-sparse, then

(1− a)(deg +q)− k ≤ ∆ + q ≤ (1 + a)(deg +q) + k.

on Cc(V ), where if (G , q) is sparse, i.e., a = 0, we may choose a ∈ (0, 1) arbitraryand

k =k

2

(1

a− a).

In the other case, i.e. a > 0, we may choose

a =

√min

(14 , a

2)

+ 2a+ a2

(1 + a)and k = max

(max

(32 ,

1a − a

)k

2(1 + a), 2k(1− a)

).

Proof. Let f ∈ Cc(V ) be complex valued. Assume first that 〈f, (deg +q)f〉 < k‖f‖2.In this case, remembering ∆ ≤ 2 deg, we can choose a ∈ (0, 1) arbitrary and k suchthat

k ≥ 2(1− a)k.

So, assume 〈f, (deg +q)f〉 ≥ k‖f‖2. Using an area and a co-area formula (cf. [KL2,Theorem 12 and Theorem 13]) with

Ωt := x ∈ V | |f(x)|2 > t,in the first step and the assumption of sparseness in the third step, we obtain

〈f,(deg +q)f〉 − k‖f‖2 =

∫ ∞

0

(deg(Ωt) + q(Ωt)− k|Ωt|

)dt

=

∫ ∞

0

(2|EΩt

|+ |∂Ωt|+ q(Ωt)− k|Ωt|)dt

≤ (1 + a)

∫ ∞

0

|∂Ωt|+ q(Ωt)dt

=(1 + a)

2

∑

x,y,x∼y

∣∣|f(x)|2 − |f(y)|2∣∣+ (1 + a)

∑

x

q(x)|f(x)|2

≤ (1 + a)

2

∑

x,y,x∼y|(f(x)− f(y))(f(x) + f(y))|+ (1 + a)

∑

x

q(x)|f(x)|2

≤ (1 + a)

2

( ∑

x,y,x∼y|f(x)− f(y)|2 + 2

∑

x

q(x)|f(x)|2)1/2

×( ∑

x,y,x∼y|f(x) + f(y)|2 + 2

∑

x

q(x)|f(x)|2)1/2

= (1 + a)〈f, (∆ + q)f〉 12(2〈f, (deg +q)f〉 − 〈f, (∆ + q)f〉

) 12 ,


where we used the Cauchy-Schwarz inequality in the last inequality and basic al-gebraic manipulation in the last equality. Since the left hand side is non-negativeby the assumption 〈f, (deg +q)f〉 ≥ k‖f‖2, we can take square roots on both sides.To shorten notation, we assume for the rest of the proof q ≡ 0 since the proof withq 6= 0 is completely analogous.

Reordering the terms, yields

(1 + a)2〈f,∆f〉2 − 2(1 + a)2〈f, deg f〉〈f,∆f〉+ (〈f, (deg−k)f〉)2 ≤ 0.

Resolving the quadratic expression above gives,

〈f, deg f〉 −√δ ≤ 〈f,∆f〉 ≤ 〈f, deg f〉+

√δ,

withδ := 〈f, deg f〉2 − (1 + a)−2(〈f, (deg−k)f〉)2.

Using 4ξζ ≤ (ξ + ζ)2, ξ, ζ ≥ 0, for all 0 < λ < 1, we estimate δ as follows

(1 + a)2δ = (2a+ a2)〈f, deg f〉2 + k‖f‖2〈f, (2 deg−k)f〉

≤ (2a+ a2)〈f, deg f〉2 +

(λ〈f, deg f〉+

k

2

(1

λ− λ)‖f‖2

)2

≤(√

λ2 + 2a+ a2〈f, deg f〉+k

2

(1

λ− λ)‖f‖2

)2

.

If a = 0, i.e., the k-sparse case, then we take λ = a to get

δ ≤ k‖f‖2〈f, 2 deg f〉 ≤(a〈f, deg f〉+

k

2

(1

a− a)‖f‖2

)2

.

As k/2a ≥ 2(1− a)k, this proves the desired inequality with k = k/2a.If a > 0, we take λ = min

(12 , a)to get

(1 + a)2δ =

((√min

(1

4, a2

)+ 2a+ a2

)〈f, deg f〉+

k

2max

(3

2,

(1

a− a))‖f‖2

)2

.

Keeping in mind the restriction k ≥ 2(1− a)k for the case 〈f, (deg +q)f〉 < k‖f‖2,this gives the statement with the choice of (a, k) in the statement of the lemma.

The two lemmas above are sufficient to prove Theorem 2.2 for the case q ≥ 0. Anapplication of Lemma A.3 turns the lower bound of Lemma 2.7 into a correspondinglower bound. This straightforward argument does not work for the upper bound.However, the following surprising lemma shows that such a lower bound by degautomatically implies the corresponding upper bound. There is a deeper reasonfor this fact which shows up in the context of magnetic Schrödinger operators.We present the non-magnetic version of the statement here for the sake of beingself-contained in this section. For the more conceptual and more general magneticversion, we refer to Lemma 4.4.

Lemma 2.8 (Upside-Down-Lemma – non-magnetic version). Let G := (V ,E ) bea graph and q : V → R. Assume there are a ∈ (0, 1), k ≥ 0 such that for allf ∈ Cc(V ),

(1− a)〈f, (deg +q)f〉 − k‖f‖2 ≤ 〈f,∆f + qf〉,


then for all f ∈ Cc(V ), we also have

〈f,∆f + qf〉 ≤ (1 + a)〈f, (deg +q)f〉+ k‖f‖2.Proof. By a direct calculation we find for f ∈ Cc(V )

〈f, (2 deg−∆)f〉 =1

2

∑

x,y∈V ,x∼y(2|f(x)|2 + 2|f(y)|2)− |f(x)− f(y)|2)

=1

2

∑

x,y,x∼y|f(x) + f(y)|2 ≥ 1

2

∑

x,y,x∼y||f(x)| − |f(y)||2

= 〈|f |,∆|f |〉.Adding q to the inequality and using the assumption gives after reordering

〈f, (∆ + q)f〉 − 2〈f, (deg +q)f〉 ≤ −〈|f |, (∆ + q)|f |〉≤ −(1− a)〈|f |, (deg +q)|f |〉+ k〈|f |, |f |〉= −(1− a)〈f, (deg +q)f〉+ k〈f, f〉

which yields the assertion. Proof of Theorem 2.2. The implication (i)⇒(iii) follows from Lemma 2.7 appliedwith q+ and from Lemma A.3 with q. The implication (iii)⇒(ii) follows fromthe Upside-Down-Lemma above. Furthermore, (ii)⇒(i) is implied by Lemma 2.5.The equivalence (ii)⇔(iv) follows from an application of the Closed Graph The-orem, Theorem A.1. Finally, the statements about discreteness of spectrum andeigenvalue asymptotics follow from an application of the Min-Max-Principle, The-orem A.2. Proof of Theorem 1.1. (a) follows from Lemma 2.7. The other statements followdirectly from Theorem 2.2.

As a corollary we can now determine the potentials in the class K0+ explicitlyand give necessary and sufficient criteria for potentials being in Kα, α ∈ (0, 1).

Corollary 2.9. Let (G , q) be an (a, k)-sparse graph for some a, k ≥ 0.(a) The potential q is in K0+ if and only if for all α ∈ (0, 1) there is κα ≥ 0

such that

q− ≤ α(deg +q+) + κα.

(b) Let α ∈ (0, 1) and a =√

min(1/4, a2) + 2a+ a2/(1 + a) (as given byLemma 2.7). If there is κα ≥ 0 such that q− ≤ α(deg +q+) + κα, thenq ∈ Kα/(1−a). On the other, hand if q ∈ Kα, then there is κα ≥ 0 suchthat q− ≤ α(1 + a)(deg +q+) + κα.

Proof. Using the assumption q− ≤ α(deg +q−) + κα and the lower bound of Theo-rem 2.2 (ii), we infer

q− ≤ α(deg +q+) + κα ≤α

(1− a)(∆ + q+) +

α

(1− a)k + κα.

Conversely, q ∈ Kα and the upper bound of Theorem 2.2 (ii) yields

q− ≤ α(∆ + q+) + Cα ≤ α(1 + a)(deg +q+) + αk + Cα.

Hence, (a) follows. For (b), notice that a =√

min(1/4, a2) + 2a+ a2/(1 + a) byLemma 2.7.


3. Almost-sparseness and asymptotic of eigenvalues

In this section we prove better estimates on the eigenvalue asymptotics in a morespecific situation. Looking at the inequality in Theorem 2.2 (ii) it seems desirableto have a = 0. As this is impossible when the degree is unbounded, we consider asequence of a that tends to 0. Keeping in mind Remark 2.3, this leads naturally tothe following definition.

Definition. Let G := (V ,E ) be a graph and q : V → R. We say (G , q) is almostsparse if for all ε > 0 there is kε ≥ 0 such that (G , q) is (ε, kε)-sparse, i.e., for anyfinite set W ⊆ V the induced subgraph GW := (W ,EW ) satisfies

2|EW | ≤ kε|W |+ ε (|∂W |+ q+(W )) .

Remark 3.1. (a) Every sparse graph G is almost sparse.(b) For an almost sparse graph (G , q), every graph (G , q′) with q′ ≥ q is almostsparse.

The main result of this section shows how the first order of the eigenvalue asymp-totics in the case of discrete spectrum can be determined for almost sparse graphs.

Theorem 3.2. Let G := (V ,E ) be a graph and q ∈ K0+ . The following assertionsare equivalent:

(i) (G , q) is almost sparse.(ii) For every ε > 0 there are kε ≥ 0 such that on Cc(V )

(1− ε)(deg +q)− kε ≤ ∆ + q ≤ (1 + ε)(deg +q) + kε.

(iii) For every ε > 0 there are kε ≥ 0 such that on Cc(V )

(1− ε)(deg +q)− kε ≤ ∆ + q.

Moreover, D((∆+q)1/2) = D((deg +q)1/2) and the operator ∆+q has purely discretespectrum if and only if lim inf |x|→∞(deg + q)(x) =∞. In this case, we have

limn→∞

λn(∆ + q)

λn(deg +q)= 1.

Proof. The statement is a direct application of Theorem 2.2 if one keeps track ofthe constants given explicitly by Lemma 2.5, Lemma 2.7 and Lemma A.3.

4. Magnetic Laplacians

In this section, we consider magnetic Schrödinger operators. Clearly, every lowerbound can be deduced from Kato’s inequality. However, for the eigenvalue asymp-totics we also need to prove an upper bound.

We fix a phase

θ : V × V → R/2πZ such that θ(x, y) = −θ(y, x).

For a potential q : V → [0,∞) we consider the magnetic Schrödinger operator∆θ + q defined as

D(∆θ + q) :=ϕ ∈ `2(V ) |

(v 7→

∑

x∼y(ϕ(x)− eiθ(x,y)ϕ(x)) + q(x)ϕ(x)

)∈ `2(V )

(∆θ + q)ϕ(x) :=∑

x∼y(ϕ(x)− eiθ(x,y)ϕ(y)) + q(x)ϕ(x).


A computation for ϕ ∈ Cc(V ) gives

〈ϕ, (∆θ + q)ϕ〉 =1

2

∑

x,y,x∼y

∣∣∣ϕ(x)− eiθ(x,y)ϕ(y)∣∣∣2

+∑

x

q(x)|ϕ(x)|2.

The operator is non-negative and selfadjoint as it is essentially selfadjoint on Cc(V )(confer e.g. [G]). For α > 0, let K θ

α be the class of real-valued potentials q suchthat q− ≤ α(∆θ + q+) + Cα for some Cα ≥ 0. Denote

K θ0+ =

⋂

α∈(0,1)

K θα .

Again, for α ∈ (0, 1) and q ∈ K θα , we define ∆θ + q to be the form sum of ∆θ + q+

and −q−.We present our result for magnetic Schrödinger operators which has one impli-

cation from the equivalences of Theorem 2.2 and Theorem 3.2.

Theorem 4.1. Let G := (V ,E ) be a graph, θ be a phase and q ∈ K θ0+ be a potential.

Assume (G , q) is (a, k)-sparse for some a, k ≥ 0. Then, we have the following:(a) There are a ∈ (0, 1), k ≥ 0 such that on Cc(V )

(1− a)(deg +q)− k ≤ ∆θ + q ≤ (1 + a)(deg +q) + k.

(b) D(|∆θ + q|1/2

)= D

(|deg + q|1/2

).

(c) The operator ∆θ + q has purely discrete spectrum if and only if

lim inf|x|→∞

(deg + q)(x) =∞.

In this case, if (G , q) is additionally almost sparse, then

lim infλ→∞

λn(∆θ + q)

λn(deg + q)= 1.

Remark 4.2. (a) The constants a and k can chosen to be the same as the oneswe obtained in the proof of Theorem 2.2, i.e., these constants are explicitly givencombining Lemma 2.7 and Lemma A.3.(b) Statement (a) and (b) of the theorem above remain true for q ∈ Kα, α ∈ (0, 1)since Kα ⊆ K θ

α by Kato’s inequality below.

We will prove the theorem by applying Theorem 2.2 and Theorem 3.2. Theconsiderations heavily rely on Kato’s inequality and a conceptual version of theUpside-Down-Lemma, Lemma 2.8, which shows that a lower bound for ∆+q impliesan upper and lower bound on ∆θ + q. Secondly, in Theorem 3.2 potentials in K0+

are considered, while here we start with the class K θ0+ . However, it can be seen

that K0+ = K θ0+ in the case of (a, k)-sparse graph, see Lemma 4.5 below.

As mentioned above a key fact is Kato’s inequality, see e.g. [DM, Lemma 2.1] or[GKS, Theorem 5.2.b].

Proposition 4.3 (Kato’s inequality). Let G := (V ,E ) be a graph, θ be a phaseand q : V → R. For all f ∈ Cc(V), we have

〈|f |, (∆|f |+ q|f |)〉 ≤ 〈f, (∆θf + qf)〉.In particular, for all α > 0

Kα ⊆ K θα and K0+ ⊆ K θ

0+ .


Proof. The proof of the inequality can be obtained by a direct calculation. Thesecond statement is an immediate consequence.

The next lemma is a rather surprising observation. It is the magnetic version ofthe Upside-Down-Lemma, Lemma 2.8.

Lemma 4.4 (Upside-Down-Lemma – magnetic version). Let G := (V ,E ) be agraph, θ be a phase and q : V → R be a potential. Assume that there are a ∈ (0, 1)

and k ≥ 0 such that for all f ∈ Cc(V ), we have

(1− a)〈f, (deg +q)f〉 − k‖f‖2 ≤ 〈f,∆f + qf〉then for all f ∈ Cc(V ), we also have

(1− a)〈f, (deg +q)f〉 − k‖f‖2 ≤ 〈f,∆θf + qf〉 ≤ (1 + a)〈f, (deg +q)f〉+ k‖f‖2.Proof. The lower bound follows directly from Kato’s inequality and the lower boundfrom the assumption (since 〈f, (deg +q)f〉 = 〈|f |, (deg +q)|f |〉 for all f ∈ Cc(V )).Now, observe that for all θ

∆θ = 2 deg−∆θ+π.

So, the upper bound for ∆θ + q follows from the lower bound of ∆θ+π + q whichwe deduced from Kato’s inequality.

The lemma above allows to relate the classes Kα and K θα for (a, k)-sparse graphs.

Lemma 4.5. For a, k ≥ 0 let G := (V ,E ) be an (a, k)-sparse graph, θ be a phaseand α > 0. Then,

K θα ⊆ Kα′ , for α′ =

1 + a

1− a α,

where a is given in Lemma 2.7. In particular,

K θ0+ = K0+ .

Moreover, if (G , q) is almost-sparse, then

K θα ⊆ Kα′ , for all α′ > α.

Proof. Let q ∈ K θα . Applying Lemma 2.7, we get ∆ + q+ ≥ (1− a)(deg +q+)− k.

Now, by the virtue of the Upside-Down-Lemma, Lemma 4.4, we infer

∆θ + q+ ≤ (1 + a)(deg +q+) + k ≤ 1 + a

1− a (∆ + q+) +2

1− a k

which implies the first statement and K θ0+ ⊆ K0+ . The reverse inclusion K θ

0+ ⊇K0+ follows from Kato’s inequality, Lemma 4.3. For almost sparse graphs a canbe chosen arbitrary small and accordingly a (from Lemma 2.7) becomes arbitrarysmall. Hence, the statement K θ

α ⊆ Kα′ , for α′ > α follows from the inequalityabove.

Proof of Theorem 4.1. Let q ∈ K θ0+ . By Lemma 4.5, q ∈ K0+ . Thus, (a) follows

from Theorem 2.2 and Lemma 4.4. Using (a) statement (b) follows from an appli-cation of the Closed Graph Theorem, Theorem A.1 and statement (c) follows froman application of the Min Max Principle, Theorem A.2.


Remark 4.6. Instead of using Kato’s inequality one can also reproduce the proofof Lemma 2.7 using the following estimate

||f(x)|2 − |f(y)|2| ≤ |(f(x)− eiθ(x,y)f(y))(f(x) + e−iθ(x,y)f(y))|.So, we infer the key estimate:

〈f,(deg +q)f〉 − k‖f‖2

≤ (1 + a)〈f, (∆θ + q)f〉 12 〈f, (∆θ+π + q)f〉 12 ,

= (1 + a)〈f, (∆θ + q)f〉 12(2〈f, (deg +q)f〉 − 〈f, (∆θ + q)f〉

) 12 .

The rest of the proof is analogous.

It can be observed that unlike in Theorem 2.2 or Theorem 3.2 we do not havean equivalence in the theorem above. A reason for this seems to be that ourdefinition of sparseness does not involve the magnetic potential. This directionshall be pursued in the future. Here, we restrict ourselves to some remarks on theperturbation theory in the context of Theorem 4.1 above.

Remark 4.7. (a) If the inequality Theorem 4.1 (a) holds for some θ, then theinequality holds with the same constants for −θ and θ ± π. This can be seen bythe fact ∆θ+π = 2 deg−∆ and 〈f,∆θf〉 = 〈f,∆−θf〉 while 〈f, deg f〉 = 〈f,deg f〉for f ∈ Cc(V ).

(b) The set of θ such that Theorem 4.1 (a) holds true for some fixed a and kis closed in the product topology, i.e., with respect to pointwise convergence. Thisfollows as 〈f,∆θnf〉 → 〈f,∆θf〉 if θn → θ, n→∞, for fixed f ∈ Cc(V ).

(c) For two phases θ and θ′ let h(x) = maxy∼x |θ(x, y)− θ′(x, y)|. By a straightforward estimate lim sup|x|→∞ h(x) = 0 implies that for every ε > 0 there is C ≥ 0such that

−εdeg−C ≤ ∆θ −∆θ′ ≤ εdeg +C

on Cc(V ). We discuss three consequences of this inequality:First of all, this inequality immediately yields that if D(∆

1/2θ ) = D(deg1/2) then

D(∆1/2θ′ ) = D(deg1/2) (by the KLMN Theorem, see e.g., [RS, Theorem X.17]) which

in turn yields equality of the form domains of ∆θ and ∆θ′ .Secondly, combining this inequality with Theorem 3.2 we obtain the following:

If lim sup|x|→∞maxy∼x |θ(x, y)| = 0 and for every ε > 0 there is kε ≥ 0 such that

(1− ε) deg−kε ≤ ∆θ ≤ (1 + ε) deg +k,ε

then the graph is almost sparse and in consequence the inequality in Theorem 4.1 (a)holds for any phase.

Thirdly, using the techniques in the proof of [G, Proposition 5.2] one shows thatthe essential spectra of ∆θ and ∆θ′ coincide. With slightly more effort and thehelp of the Kuroda-Birman Theorem, [RS, Theorem XI.9] one can show that ifh ∈ `1(V ), then even the absolutely continuous spectra of ∆θ and ∆θ′ coincide.

5. Isoperimetric estimates and sparseness

In this section we relate the concept of sparseness with the concept of isoperi-metric estimates. First, we present a result which should be viewed in the light ofTheorem 2.2 as it points out in which sense isoperimetric estimates are stronger


than our notions of sparseness. In the second subsection, we present a result re-lated to Theorem 3.2. Finally, we present a concrete comparison of sparseness andisoperimetric estimates. As this section is of a more geometric flavor we restrictourselves to the case of potentials q : V → [0,∞).

5.1. Isoperimetric estimates. Let U ⊆ V and define the Cheeger or isoperimet-ric constant of U by

αU := infW ⊂U finite

|∂W |+ q(W )

deg(W ) + q(W ).

In the case where deg(W ) + q(W ) = 0, for instance when W is an isolated point,by convention the above quotient is set to be equal to 0, Note that αU ∈ [0, 1).

The following theorem illustrates in which sense positivity of the Cheeger con-stant is linked with (a, 0)-sparseness. We refer to Theorem 5.4 for precise constants.

Theorem 5.1. Given G := (V ,E ) a graph and q : V → [0,∞). The followingassertions are equivalent

(i) αV > 0.(ii) There is a ∈ (0, 1)

(1− a)(deg +q) ≤ ∆ + q ≤ (1 + a)(deg +q).

(iii) There is a ∈ (0, 1) such that

(1− a)(deg +q) ≤ ∆ + q.

The implication (iii)⇒(i) is already found in [G, Proposition 3.4]. The implica-tion (i)⇒(ii) is a consequence from standard isoperimetric estimates which can beextracted from the proof of [KL2, Proposition 15].

Proposition 5.2 ([KL2]). Let G := (E ,V ) be a graph and q : V → [0,∞). Then,for all U ⊆ V we have on Cc(U ).

(1−

√1− α2

U

)(deg +q) ≤ ∆ + q ≤

(1 +

√1− α2

U

)(deg +q).

5.2. Isoperimetric estimates at infinity. Let the Cheeger constant at infinitybe defined as

α∞ = supK ⊆V finite

αV \K .

Clearly, 0 ≤ αV ≤ αU ≤ α∞ ≤ 1 for any U ⊆ V .As a consequence of Proposition 5.2, we get the following theorem.

Theorem 5.3. Let G := (E ,V ) be a graph and q : V → [0,∞) be a potential.Assume α∞ > 0. Then, we have the following:

(a) For every ε > 0 there is kε ≥ 0 such that on Cc(V )

(1− ε)(1−

√1− α2∞

)(deg +q)− kε ≤ ∆ + q

≤ (1 + ε)(1 +

√1− α2∞

)(deg +q) + kε.

(b) D((∆ + q)1/2) = D((deg +q)1/2).(c) The operator ∆ + q has purely discrete spectrum if and only if we have

lim inf |x|→∞(deg + q)(x) =∞. In this case, if additionally α∞ = 1, we get

lim infλ→∞

λn(∆ + q)

λn(deg + q)= 1.


Proof. (a) Let ε > 0 and K ⊆ V be finite and large enough such that

(1− ε)(1−

√1− α2∞

)≤(1−

√1− α2

V \K)

(1 +

√1− α2

V \K)≤ (1 + ε)

(1 +

√1− α2∞

).

From Proposition 5.2 we conclude on Cc(V \K )

(1− ε)(1−

√1− α2∞

)(deg +q) ≤

(1−

√1− α2

V \K)(deg +q)

≤ ∆ + q ≤(1 +

√1− α2

V \K)(deg +q)

≤ (1 + ε)(1 +

√1− α2∞

)(deg +q)

By local finiteness the operators 1V \K (∆ + q)1V \K and 1V \K (deg +q)1V \K arebounded (indeed, finite rank) perturbations of ∆ + q and deg +q. This gives riseto the constants kε and the inequality of (a) follows. Now, (b) is an immediateconsequence of (a), and (c) follows by the Min-Max-Principle, Theorem A.2.

5.3. Relating sparseness and isoperimetric estimates. We now explain howthe notions of sparseness and isoperimetric estimates are exactly related.

First, we consider classical isoperimetric estimates.

Theorem 5.4. Let G := (V ,E ) be a graph, a, k ≥ 0, and let q : V → [0,∞) be apotential.

(a) αV ≥1

1 + aif and only if (G , q) is (a, 0)-sparse.

(b) If (G , q) is (a, k)-sparse, then

αV ≥d− kd(1 + a)

,

where d := infx∈V (deg +q)(x). In particular, αV > 0 if d > k.(c) Suppose that (G , q) is (a, k)-sparse graph that is not (a, k′)-sparse for all

k′ < k. Suppose also that there is d such that d = deg(x) + q(x) for allx ∈ V . Then

αV =d− kd(1 + a)

.

Proof. Let W ⊂ V be a finite set. Recalling the identity deg(W ) = 2|EW |+ |∂W |we notice that

1

1 + a≤ |∂W |+ q(W )

(deg +q)(W )

is equivalent to

2|EW | ≤ a(|∂W |+ q(W ))

which proves (a).For (b), the definition of (a, k)-sparseness yields

|∂W |+ q(W )

(deg +q)(W )= 1− 2|EW |

(deg +q)(W )≥ 1− a |∂W |+ q(W )

(deg +q)(W )− k |W |

(deg +q)(W ).

This concludes immediately.


For (c), the lower bound of αV follows from (b). Since (G , q) is not (a, k′)-sparsethere is a finite W0 ⊂ V such that

|∂W0|+ q(W0)

(deg +q)(W0)< 1− a |∂W0|+ q(W0)

(deg +q)(W0)− k′ |W0|

(deg +q)(W0).

Therefore αV < (d− k′)/(d(1 + a)).

We next address the relation between almost sparseness and isoperimetry andshow two “almost equivalences”.

Theorem 5.5. Let G := (V ,E ) be a graph and let q : V → [0,∞) be a potential.(a) If α∞ > 0, then (G , q) is (a, k)-sparse for some a > 0, k ≥ 0. On the other

hand, if (G , q) is (a, k)-sparse for some a > 0, k ≥ 0 and

l := lim inf|x|→∞

(deg +q)(x) > k,

then

α∞ ≥l − kl(a+ 1)

> 0,

if l is finite and α∞ ≥ 1/(1 + a) otherwise.(b) If α∞ = 1, then (G , q) is almost sparse. On the other hand, if (G , q) is

almost sparse and lim inf |x|→∞(deg +q)(x) =∞, then α∞ = 1.

Proof. The first implication of (a) follows from Theorem 5.3 (a) and Theorem 2.2(ii)⇒(i). For the opposite direction let ε > 0 and K ⊆ V be finite such thatdeg +q ≥ l − ε on V \K . Using the formula in the proof of Theorem 5.4 above,yields for W ⊆ V \K

|∂W |+ q(W )

(deg +q)(W )= 1− 2|EW |

(deg +q)(W )≥ 1− k|W |+ a(|∂W |+ q(W ))

(deg +q)(W )

≥ 1− k

(l − ε) −a(|∂W |+ q(W ))

(deg +q)(W ).

This proves (a).The first implication of (b) follows from Theorem 5.3 (a) and Theorem 3.2 (ii)⇒(i).The other implication follows from (a) using the definition of almost sparseness.

Remark 5.6. (a) We point out that without the assumptions on (deg +q) the con-verse implications do not hold. For example the Cayley graph of Z is 2-sparse (cf.Lemma 6.2), but has α∞ = 0.(b) Observe that α∞ = 1 implies lim inf |x|→∞(deg +q)(x) = ∞. Hence, (b) canbe rephrased as the following equivalence: α∞ = 1 is equivalent to (G , q) almostsparse and lim inf |x|→∞(deg +q)(x) =∞.

The previous theorems provides a slightly simplified proof of [K1] which alsoappeared morally in somewhat different forms in [D1, Woe].

Corollary 5.7. Let G := (V ,E ) be a planar graph.(a) If for all vertices deg ≥ 7, then αV > 0.(b) If for all vertices away from a finite set deg ≥ 7, then α∞ > 0.

Proof. Combine Theorem 5.4 and Theorem 5.5 with Lemma 6.2.


6. Examples

6.1. Examples of sparse graphs. To start off, we exhibit two classes of sparsegraphs. First we consider the case of graphs with bounded degree.

Lemma 6.1. Let G := (V ,E ) be a graph. Assume D := supx∈V deg(x) < +∞,then G is D-sparse.

Proof. Let W be a finite subset of V . Then, 2|EW | ≤ deg(W ) ≤ D|W |.

We turn to graphs which admit a 2-cell embedding into Sg, where Sg denotesa compact orientable topological surface of genus g. (The surface Sg might bepictured as a sphere with g handles.) Admitting a 2-cell embedding means thatthe graphs can be embedded into Sg without self-intersection. By definition we saythat a graph is planar when g = 0. Note that unlike other possible definitions ofplanarity, we do not impose any local compactness on the embedding.

Lemma 6.2. (a) Trees are 2-sparse.(b) Planar graphs are 6-sparse.(c) Graphs admitting a 2-cell embedding into Sg with g ≥ 1 are 4g + 2-sparse.

Proof. (a) Let G := (V ,E ) be a tree and GW := (W ,EW ) be a finite inducedsubgraph of G . Clearly |EW | ≤ |W | − 1. Therefore, every tree is 2-sparse.

We treat the cases (b) and (c) simultaneously. Let G := (V ,E ) be a graph whichis connected 2-cell embedded in Sg with g ≥ 0 (as remarked above planar graphscorrespond to g = 0). Let GW := (W ,EW ) be a finite induced subgraph of G which,clearly, also admits a 2-cell embedding into Sg. The statement is clear for |W | ≤ 2.Assume |W | ≥ 3. Let FW be the faces induced by GW := (W ,EW ) in Sg. Here,all faces (even the outer one) contain at least 3 edges, each edge belongs only to 2faces, thus,

2|EW | ≥ 3|FW |.Euler’s formula, |W | − |EW |+ |FW | = 2− 2g, gives then

2− 2g + |EW | = |W |+ |FW | ≤ |W |+2

3|EW |

that is|EW | ≤ 3|W |+ 6(g − 1) ≤ max(2g + 1, 3)|W |.

This concludes the proof.

Next, we explain how to construct sparse graphs from existing sparse graphs.

Lemma 6.3. Let G1 := (V1,E1) and G2 := (V2,E2) be two graphs.(a) Assume V1 = V2, G1 is k1-sparse and G2 is k2-sparse. Then, G := (V ,E )

with E := max(E1,E2) is (k1 + k2)-sparse.(b) Assume G1 is k1-sparse and G2 is k2-sparse. Then G1 ⊕ G2 := (V ,E ) with

where V := V1 × V2 and

E ((x1, x2), (y1, y2)) := δx1(y1) · E2(x2, y2) + δx2(y2) · E1(x1, y1),

is (k1 + k2)-sparse.(c) Assume V1 = V2, G1 is k-sparse and E2 ≤ E1. Then, G2 is k-sparse.


Proof. For (a) let W ⊆ V be finite and note that |EW | ≤ |E1,W | + |E2,W |. For (b)let p1, p2 the canonical projections from V to V1 and V2. For finite W ⊆ V weobserve

|EW | = |E1,p1(W )|+ |E2,p2(W )| ≤ k1|p2(W )|+ k2|p1(W )| ≤ (k1 + k2)|W |.

For (c) and W ⊆ V finite, we have |E2,W | ≤ |E1,W | which yields the statement.

Remark 6.4. (a) We point out that there are bi-partite graphs which are notsparse. See for example [G, Proposition 4.11] or take an antitree, confer [KLW,Section 6], where the number of vertices in the spheres grows monotonously to ∞.(b) The last point of the lemma states that the k-sparseness is non-decreasing whenwe remove edges from the graph. This is not the case for the isoperimetric constant.

6.2. Examples of almost-sparse and (a, k)-sparse graph. We construct a se-ries of examples which are perturbations of a radial tree. They illustrate thatsparseness, almost sparseness and (a, k)-sparseness are indeed different concepts.

Let β = (βn), γ = (γn) be two sequences of natural numbers. Let T = T (β)with T = (V ,E T ) be a radial tree with root o and vertex degree βn at the n-thsphere, that is every vertex which has natural graph distance n to o has (βn − 1)forward neighbors. We denote the distance spheres by Sn. We let G (β, γ) be theset of graphs G := (V ,E G ) that are super graphs of T such that the inducedsubgraphs GSn are γn-regular and E G (x, y) = E T (x, y) for x ∈ Sn, y ∈ Sm, m 6= n.

Observe that G (β, γ) is non empty if and only if γn∏nj=0(βj − 1) is even and

γn < |Sn| =∏nj=0(βj − 1) for all n ≥ 0.

Figure 1. G with β = (3, 3, 4, . . .) and γ = (0, 2, 4, 5, . . .).

Proposition 6.5. Let β, γ ∈ NN00 , a = lim supn→∞ γn/βn and G ∈ G (β, γ).

(a) If a = 0, then G is almost sparse. The graph G is sparse if and only iflim supn→∞ γn <∞.

(b) If a > 0, then G is (a′, k)-sparse for some k ≥ 0 if a′ > a. Conversely, ifG is (a′, k)-sparse for some k ≥ 0, then a′ ≥ a.

Proof. Let ε > 0 and let N ≥ 0 be so large that

γn ≤ (a+ ε)βn, n ≥ N.


Set Cε :=∑N−1n=0 degG (Sn). Let W be a non-empty finite subset of V . We calculate

2|E GW |+ |∂G W | = degG (W ) = degT (W ) +

∑

n≥0

|W ∩ Sn|γn

≤ degT (W ) + (a+ ε)∑

n≥0

|W ∩ Sn|βn +

N−1∑

n=0

|W ∩ Sn|γn

≤ (1 + a+ ε) degT (W ) + Cε|W |= 2(1 + a+ ε)|E T

W |+ (1 + a+ ε)|∂T W |+ Cε|W |≤ (2(1 + a+ ε) + Cε)|W |+ (1 + a+ ε)|∂T W |,

where we used that trees are 2-sparse in the last inequality. Finally, since |∂G W | ≥|∂T W |, we conclude

2|E GW | ≤ (2(1 + a+ ε) + Cε) |W |+ (a+ ε)|∂G W |.

This shows that the graph in (a) with a = 0 is almost sparse and that the graph in(b) with a > 0 is (a+ ε, kε)-sparse for ε > 0 and kε = 2(1 + a+ ε) +Cε. Moreover,for the other statement of (a) let k0 = lim supn→∞ γn and note that for GSn

2|ESn| = γn|Sn|.

Hence, if k0 = ∞, then G is not sparse. On the other hand, if k0 < ∞, then G is(k0 + 2)-sparse by Lemma 6.3 as T is 2-sparse by Lemma 6.2. This finishes theproof of (a). Finally, assume that G is (a′, k)-sparse with k ≥ 0. Then, for W = Sn

γn|Sn| = 2|ESn| ≤ k|Sn|+ a′|∂GSn| = k|Sn|+ a′βn|Sn|

Dividing by βn|Sn| and taking the limit yields a ≤ a′. This proves (b).

Remark 6.6. In (a), we may suppose alternatively that we have the completegraph on Sn and the following exponential growth limn→∞

|Sn||Sn+1| = 0.

Appendix A. Some general operator theory

We collect some consequences of standard results from functional analysis thatare used in the paper. Let H be a Hilbert space with norm ‖ · ‖. For a quadraticform Q, denote the form norm by ‖ · ‖Q :=

√Q(·) + ‖ · ‖2. The following is a direct

consequence of the Closed Graph Theorem, (confer e.g. [We, Satz 4.7]).

Theorem A.1. Let (Q1,D(Q1)) and (Q2,D(Q2)) be closed non-negative quadraticforms with a common form core D0. Then, the following are equivalent:

(i) D(Q1) ≤ D(Q2).(ii) There are constants c1 > 0, c2 ≥ 0 such that c1Q2 − c2 ≤ Q1 on D0.

Proof. If (ii) holds, then any ‖ · ‖Q1-Cauchy sequence is a ‖ · ‖Q2

-Cauchy sequence.Thus, (ii) implies (i). On the other hand, consider the identity map j : (D(Q1), ‖ ·‖Q1)→ (D(Q2), ‖ · ‖Q2). The map j is closed as it is defined on the whole Hilbertspace (D(Q1), ‖ · ‖Q1) and, thus, bounded by the Closed Graph Theorem [RS,Theorem III.12] which implies (i).

For a selfadjoint operator T which is bounded from below, we denote the bottomof the spectrum by λ0(T ) and the bottom of the essential spectrum by λess

0 (T ). Letn(T ) ∈ N0 ∪ ∞ be the dimension of the range of the spectral projection of


(−∞, λess0 (T )). For λ0(T ) < λess

0 (T ) we denote the eigenvalues below λess0 (T ) by

λn(T ), for 0 ≤ n ≤ n(T ), in increasing order counted with multiplicity.

Theorem A.2. Let (Q1,D(Q1)) and (Q2,D(Q2)) be closed non-negative quadraticforms with a common form core D0 and let T1 and T2 be the corresponding selfad-joint operators. Assume there are constants c1 > 0, c2 ∈ R such that on D0

c1Q2 − c2 ≤ Q1.

Then, c1λn(T2)−c2 ≤ λn(T1), for 0 ≤ n ≤ min(n(T1), n(T2)). Moreover, c1λess0 (T2)−

c2 ≤ λess0 (T1), in particular, σess(T1) = ∅ if σess(T2) = ∅ and in this case

c1 ≤ lim infn→∞

λn(T1)

λn(T2).

Proof. Letting

µn(T ) = supϕ1,...,ϕn∈H

inf06=ψ∈ϕ1,...,ϕn⊥∩D0

〈Tψ, ψ〉〈ψ,ψ〉 ,

for a selfadjoint operator T , we know by the Min-Max-Principle [RS, ChapterXIII.1] µn(T ) = λn(T ) if λn(T ) < λess

0 (T ) and µn(T ) = λess0 (T ) otherwise, n ≥ 0.

Assume n ≤ minn(T1), n(T2) and let ϕ(j)0 , . . . , ϕ

(j)n be the eigenfunctions of Tj to

λ0(Tj), . . . , λn(Tj) we get

c1λn(T2)− c3 = inf06=ψ∈ϕ(2)

1 ,...,ϕ(2)n ⊥∩D0

(c1〈T2ψ,ψ〉〈ψ,ψ〉 − c3

)

≤ inf06=ψ∈ϕ(2)

1 ,...,ϕ(2)n ⊥∩D0

〈T1ψ,ψ〉〈ψ,ψ〉 ≤ µn(T1) = λn(T1)

This directly implies the first statement. By a similar argument the statementabout the bottom of the essential spectrum follows, in particular, λess

0 (T2) = ∞implies limn→∞ µn(T1) = ∞ and, thus, λess

0 (T1) = ∞. In this case λn(T2) → ∞,n→∞, which implies the final statement.

Finally, we give a lemma which helps us to transform inequalities under formperturbations.

Lemma A.3. Let (Q1,D(Q1)), (Q2,D(Q2)) and (q,D(q)) be closed symmetricnon-negative quadratic forms with a common form core D0 such that there areα ∈ (0, 1), Cα ≥ 0 such that

q ≤ αQ1 + C

on D0. If for a ∈ (0, 1) and k ≥ 0

(1− a)Q2 − k ≤ Q1 on D0,

then(1− α)(1− a)

(1− α(1− a))(Q2 − q)−

(1− α)k + aCα(1− α(1− a))

≤ Q1 − q, on D0.

In particular, if a→ 0+, then (1− α)(1− a)/(1− α(1− a))→ 1− and if α→ 0+,then (1− α)(1− a)/(1− α(1− a))→ (1− a).


Proof. The assumption on q implies

q ≤ α

(1− α)(Q1 − q) +

Cα(1− α)

.

We subtract (1−a)q on each side of the lower bound in (1−a)Q2−k ≤ Q1. Then,we get

(1− a)(Q2 − q)− k ≤ (Q1 − q) + aq ≤ 1− α(1− a)

(1− α)(Q1 − q) +

aCα(1− α)

and, thus, the asserted inequality follows.

Acknowledgement. MB was partially supported by the ANR project HAB(ANR-12-BS01-0013-01). SG was partially supported by the ANR project GeRaSicand SQFT. MK enjoyed the hospitality of Bordeaux University when this workstarted. Moreover, MK acknowledges the financial support of the German ScienceFoundation (DFG), Golda Meir Fellowship, the Israel Science Foundation (grantno. 1105/10 and no. 225/10) and BSF grant no. 2010214.

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(1996), 109–138.[We] J. Weidmann, Lineare Operatoren in Hilberträumen 1, B. G. Teubner, Stuttgart, 2000.[Woe] W. Woess, A note on tilings and strong isoperimetric inequality, Math. Proc. Camb.

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Math. J. 58 (2009), 1419–1441.

Michel Bonnefont, Institut de Mathématiques de Bordeaux Université Bordeaux 1351, cours de la Libération F-33405 Talence cedex, France

E-mail address: [email protected]

Sylvain Golénia, Institut de Mathématiques de Bordeaux Université Bordeaux 1351, cours de la Libération F-33405 Talence cedex, France


Matthias Keller, Friedrich Schiller Universität Jena, Mathematisches Institut,07745 Jena, Germany


CHAPTER 8

B. Hua, M. Keller, Harmonic functions of generalgraph Laplacians, Calculus of Variations and

Partial Differential Equations 51 (2014), 343–362.

193

Calc. Var. (2014) 51:343–362DOI 10.1007/s00526-013-0677-6 Calculus of Variations

Harmonic functions of general graph Laplacians

Bobo Hua · Matthias Keller

Received: 10 April 2013 / Accepted: 5 September 2013 / Published online: 1 October 2013© Springer-Verlag Berlin Heidelberg 2013

Abstract We study harmonic functions on general weighted graphs which allow for a com-patible intrinsic metric. We prove an L p Liouville type theorem which is a quantitativeintegral L p estimate of harmonic functions analogous to Karp’s theorem for Riemannianmanifolds. As corollaries we obtain Yau’s L p-Liouville type theorem on graphs, identify thedomain of the generator of the semigroup on L p and get a criterion for recurrence. As a sideproduct, we show an analogue of Yau’s L p Caccioppoli inequality. Furthermore, we derivevarious Liouville type results for harmonic functions on graphs and harmonic maps fromgraphs into Hadamard spaces.

Mathematics Subject Classification 31C05 · 58E20

1 Introduction

The study of harmonic functions is a fundamental topic in various areas of mathematics.An important question is which subspaces of harmonic functions are trivial, that is, theycontain only constant functions. Such results are referred to as Liouville type theorems.In Riemannian geometry L p-Liouville type theorems for harmonic functions were stud-ied for example by Yau [58], Karp [32], Li–Schoen [41] and many others. Karp’s crite-rion was later generalized by Sturm [49] to the setting of strongly local regular Dirich-let forms. Over the years there were several attempts to realize an analogous theoremfor graphs, see Holopainen–Soardi [25], Rigoli–Salvatori–Vignati [46], Masamune [44]

Communicated by J. Jost.

B. Hua (B)Max Planck Institute for Mathematics in the Sciences, 04103 Leipzig, Germanye-mail: [email protected]

M. KellerEinstein Institute of Mathematics, The Hebrew University of Jerusalem, 91904 Jerusalem, Israele-mail: [email protected]

123

344 B. Hua, M. Keller

and most recently Hua–Jost [20]. In all these works normalized Laplacians were stud-ied (often with further restrictions on the vertex degree) and certain criteria, all weakerthan Karp’s integral estimate, were obtained. The main challenge when considering graphsis the non-existence of a chain rule and, moreover, the fact that for unbounded graphLaplacians the natural graph distance is very often not the proper analogue to the Rie-mannian distance in manifolds. In this paper, we use the newly developed concept ofintrinsic metrics on graphs to prove an analogue to Karp’s theorem for general Lapla-cians on weighted graphs. Thus, we generalize all earlier results on graphs not only withrespect to the generality of the setting but also by recovering the precise analogue of Karp’scriterion.

Harmonic maps are very important nonlinear objects in geometric analysis studied thor-oughly by many authors (e.g. Eells–Sampson [11], Schoen–Yau [53], Hildebrandt–Jost–Widman [21]). In this paper, we adopt a definition of harmonic maps between metric measurespaces introduced by Jost [27–30]. In particular, we study harmonic maps from graphs intoHadamard spaces (i.e. globally non-positively curved spaces, also called CAT(0)-spaces),studied also by [26,31,39], and prove Liouville type theorems in this context. For variousLiouville theorems on manifolds, we refer to [6,9,34,54] and references therein. We provethe finite-energy Liouville theorem for harmonic maps from graphs into Hadamard spacesanalogous to the one in Cheng–Tam–Wang [9] on manifolds.

In what follows we first state and discuss our results and refer for details and precisedefinitions to Sect. 2. Our framework are weighted graphs over a discrete measure space(X, m) introduced in [35] which includes non locally finite graphs, (see also [48]). In thissetting a pseudo metric is called intrinsic if the energy measures of distance functions canbe estimated by the measure of the graph (see Definition 2.2). We further call such a pseudometric compatible if it has finite jump size and the weighted vertex degree is bounded oneach distance ball (see Definition 2.3). As the boundedness of the weighted vertex degreeis implied by finiteness of distance balls which is equivalent to metric completeness in thecase of a path metric on a locally finite graph, see [24, Theorem A.1], this assumption canbe seen as an analogue of completeness in the Riemannian manifold case. Similarly, Sturm[49] asks for precompactness of balls.

Our first main result is the following analogue to Karp’s L p Liouville theorem [32, Theo-rem 2.2], whose proof is given in Sect. 3.2. A function is called (sub)harmonic if it is in thedomain of the formal Laplacian and the formal Laplacian applied to this function is pointwise(less than or) equal to zero, (see Definition 2.1). We denote by 1Br the characteristic functionof the balls Br , r ≥ 0, which are taken with respect to an intrinsic metric about a fixed vertexo ∈ X .

Theorem 1.1 (Karp’s L p Liouville theorem) Assume a connected weighted graph allows fora compatible intrinsic metric. Then every non-negative subharmonic function f satisfying

infr0>0

∞∫

r0

r

‖ f 1Br ‖pp

dr = ∞,

for some p ∈ (1,∞), is constant.

Clearly, the integral in the theorem above diverges, whenever 0 = f ∈ L p(X, m). Thus,as an immediate corollary, we get Yau’s L p Liouville type theorem [58].

Corollary 1.2 (Yau’s L p Liouville theorem) Assume a connected weighted graph Xallows for a compatible intrinsic metric. Then every non-negative subharmonic functionin L p(X, m), p ∈ (1,∞), is constant.

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Harmonic functions of general graph Laplacians 345

Remark 1.3 (a) The results above imply the corresponding statements for harmonic func-tions by the simple observation that f+, f− and | f | of a harmonic function f are non-negative and subharmonic.

(b) Harmonicity of a function is independent of the choice of the measure m. Hence, forany non-constant harmonic function f on X, we may find a sufficiently small measurem such that f ∈ L p(X, m) for any p ∈ (0,∞), see [44]. Our theorem states that ifwe impose the restriction of compatibility on the measure and the metric, then the L p

Liouville theorem holds for 1 < p < ∞.

(c) Theorem 1.1 generalizes all earlier results on graphs [20,25,44,46] for the case p ∈(1,∞). Not only that our setting is more general—as the natural graph distance is alwaysa compatible intrinsic metric to the normalized Laplacian—but also our criterion is moregeneral. In particular, if f satisfies

lim supr→∞

1

r2 log r‖ f 1Br ‖p

p < ∞,

then the integral in Theorem 1.1 diverges. Thus, Theorem 1.1 is stronger than [20,Theorem 1.1] (which had only r2 rather than r2 log r in the denominator). The authorsof [20] observed that for the normalized Laplacian the case p ∈ (1, 2] can already beobtained by their techniques, (see [20, Remark 3.3]). Here, the missing cases p ∈ (2,∞)

are treated by adopting a subtle lemma in [25]. Moreover, our techniques would also allowa statement such as [20, Theorem 1.1] for the cases p < 1.

(d) In [35] discrete measure spaces (X, m) with the assumption that every infinite path hasinfinite measure are discussed (this assumption is denoted by (A) in [35]). It is not hardto see that for connected graphs over (X, m) every non-negative subharmonic functionL p(X, m), p ∈ [1,∞), is trivial. In fact, from every non-constant positive subharmonicfunction we can extract a sequence of vertices such that the function values increasealong this sequence (compare [35, Lemma 3.2 and Theorem 8]). Since this path hasinfinite measure, the function is not contained in L p(X, m), p ∈ [1,∞). Thus, the onlyinteresting measure spaces are those that contain an infinite path of finite measure.

(e) Sturm [49] proves an analogue for Karp’s theorem for weakly subharmonic functions.This might seem stronger, however, in our setting on graphs weak solutions of equationsare automatically solutions, [23, Theorem 2.2 and Corollary 2.3].

Corollary 1.2 allows us to explicitly determine the domain of the generator L p of thesemigroup on L p(X, m). We denote by the formal Laplacian with formal domain F . (Fordefinitions see Sect. 2.2). The proof of the corollary below is given in Sect. 3.4.

Corollary 1.4 (Domain of the L p generators) Assume a connected weighted graph X allowsfor a compatible intrinsic metric. Then, for p ∈ (1,∞), the generator L p is a restriction of and

D(L p) = u ∈ L p(X, m) ∩ F | u ∈ L p(X, m).Remark 1.5 (a) The corollary above generalizes [24, Theorem 1] to the case p ∈ (1,∞)

and settles the question in [24, Remark 3.6]. Moreover, it complements [35, Theorem 5].(b) It would be interesting to know whether there is a Liouville type theorem for functions

in D(L p) without the assumption of compatibility on the metric.

We get furthermore a sufficient criterion for recurrence analogous to [32, Theorem 3.5]and [49, Theorem 3] which generalizes for example [10, Theorem 2.2], [46, Corollary B],

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[55, Lemma 3.12], [18, Corollary 1.4], [45, Theorem 1.2] on graphs. For a characterizationof recurrence see Proposition 3.3 in Sect. 3.4, where also the proof of the corollary below isgiven.

Corollary 1.6 (Recurrence) Assume a connected weighted graph allows for a compatibleintrinsic metric. If

∞∫

1

r

m(Br )dr = ∞,

then the graph is recurrent.

Contrary to the normalized Laplacian, [20, Theorem 1.2], there is no L1 Liouville typetheorem in the general case. However, for stochastic complete graphs (see Sect. 4) we havethe following analogue to [16, Theorem 3], [49, Theorem 2]. The proof following [17] isgiven in Sect. 4. We also give counter-examples to L1 Liouville theorem which complementthe counter-examples from manifolds, [8,41].

Theorem 1.7 (Grigor’yan’s L1 theorem) Assume a connected graph X is stochastically com-plete. Then, every non-negative superharmonic function in L1(X, m) is constant.

For vertices x, y ∈ X that are connected by an edge, we denote a directed edge by xy andthe positive symmetric edge weight by μxy . We define

∇xy f = f (x) − f (y).

The following L p Caccioppoli-type inequality is a side product of our analysis. Suchan inequality was proven in [20,25,46] for bounded operators. The classical Caccioppoliinequality is the case p = 2, which can be found for graphs in [5,24,43].

Theorem 1.8 (Caccioppoli-type inequality) Assume a connected weighted graph allows fora compatible intrinsic metric and p ∈ (1,∞). Then, there is C > 0 such that for everynon-negative subharmonic function f and all 0 < r < R − 3s

∑x,y∈Br

μxy( f (x) ∨ f (y))p−2|∇xy f |2 ≤ C(R−r)2 ‖ f 1BR\Br ‖p

p,

where s is the jump size of the intrinsic metric (see Sect. 2.3).

Remark 1.9 (a) The theorem above allows for a direct proof of Corollary 1.2, confer [20,Corollary 3.1].

(b) For p ≥ 2, we can strengthen the inequality by replacing ( f (x) ∨ f (y))p−2 on the lefthand side by f p−2(x) + f p−2(y), see Remark 3.2 in Sect. 3.3, where the theorem isproven.

The following quantitative consequence of Theorem 1.1 which is a generalization ofCorollary 1.2 has various corollaries that are stated and proven in Sect. 5. For an intrinsicmetric ρ and a fixed vertex o ∈ X let

ρ1 = 1 ∨ ρ(·, o).

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Theorem 1.10 Assume a connected weighted graph X allows for a compatible intrinsicmetric ρ. If a non-negative subharmonic function f satisfies

f ∈ L p(X, mρ−21 ),

for some p ∈ (1,∞), then f is constant.

Next, we turn to harmonic maps from graphs into Hadamard spaces, (see Sect. 6, inparticular Definition 6.1). We prove the following consequence of Karp’s theorem in Sect. 6.

Theorem 1.11 (Karp’s theorem for harmonic maps) Assume a connected weighted graphX allows for a compatible intrinsic metric ρ. Let u be a harmonic map into an Hadamardspace (Y, d). If there are p ∈ (1,∞) and y ∈ Y such that

d(u(·), y) ∈ L p(X, mρ−21 ),

then u is bounded. Moreover, if mρ−21 (X) = ∞ or y is in the image of u, then u is constant.

Finally, we turn to harmonic functions and maps of finite energy, (for definitions seeSects. 2.2 and 6.2). The two theorems below stand in close relationship to the celebratedtheorem of Kendall [33, Theorem 6], (confer [22,40]). Our first result in this line is a directconsequence of Theorem 6.3 and it is an analogue to Cheng–Tam–Wang [9, Theorem 3.1].

Theorem 1.12 Assume that on a graph every harmonic function of finite energy is bounded.Then, every harmonic map from the graph into an Hadamard space is bounded.

The second result in this line is an analogue to [9, Theorem 3.2]. An Hadamard space iscalled locally compact if for any point there exists a precompact neighborhood.

Theorem 1.13 Assume that on a graph every bounded harmonic function is constant. Then,every finite-energy harmonic map from the graph into a locally compact Hadamard space isconstant.

The paper is organized as follows. In the next section, we introduce the involved conceptsand recall some basic inequalities. Section 3 is devoted to the proofs of Theorems 1.1, 1.8and the corollaries above. The proof of Theorem 1.7 and counter-examples to an L1-Liouvilletype statement are given in Sect. 4. In Sect. 5 we prove Theorem 1.10 and derive variouscorollaries. Harmonic maps from graphs into Hadamard spaces are discussed in Sect. 6.Theorem 1.11 is proven in Sect. 6.1 and Theorems 1.12 and 1.13 are proven in Sect. 6.2.Several applications are discussed in Sect. 6.3.

Throughout this paper C always denotes a constant that might change from line to line.Moreover, we use the convention that ∞ · 0 = 0, (which only appears in expressions suchas f −q(x)∇xy f with f (x) = f (y) = 0 and q > 0).

2 Set-up and preliminaries

2.1 Weighted graphs

Let X be a countable discrete set and m : X → (0,∞). Extending m additively to sets,(X, m) becomes a measure space with a measure of full support. A graph over (X, m) is

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induced by an edge weight function μ : X × X → [0,∞), (x, y) → μxy that is symmetric,has zero diagonal and satisfies

∑y∈X

μxy < ∞, x ∈ X.

If μxy > 0 we write x ∼ y and let xy and yx be the oriented edges of the graph. We writexy ⊂ A for a set A ⊆ X if both of the vertices of the edge xy are contained in A, i.e.,x, y ∈ A. When we fix an orientation for the edges we denote the directed edges often by e.

We refer to the triple (X, μ, m) as a weighted graph. We assume the graph is connected,that is for every two vertices x, y ∈ X there is a path x = x0 ∼ x1 ∼ · · · ∼ xn = y.

The spaces L p(X, m), p ∈ [1,∞), and L∞(X) are defined in the natural way. Forp ∈ [1,∞), let p∗ be its Hölder dual, i.e., 1

p + 1p∗ = 1.

2.2 Laplacians and (sub)harmonic functions

We define the formal Laplacian on the formal domain

F(X) =⎧⎨⎩ f : X → R |

∑y∈X

μxy | f (y)| < ∞ for all x ∈ X

⎫⎬⎭ ,

by

f (x) = 1

m(x)

∑y∈X

μxy( f (x) − f (y)).

Definition 2.1 (Harmonic function) A function f : X → R is called harmonic (subhar-monic, superharmonic) if f ∈ F(X) and f = 0, ( f ≤ 0, f ≥ 0).

Obviously, the measure does not play a role in the definition of harmonicity. We denoteby L the positive selfadjoint restriction of on L2(X, m) which arises from the closure Qof the restriction of the quadratic form E : X → R → [0,∞]

E( f ) = 1

2

∑x,y∈X

μxy |∇xy f |2

to Cc(X), the space of finitely supported functions, (for details see [35]). Since Q is a Dirichletform, the semigroup e−t L , t ≥ 0, extends to a C0-semigroup on L p(X, m), p ∈ [1,∞) (resp.a weak C0-semigroup for p = ∞). We denote the generators of these semigroups by L p ,p ∈ [1,∞). Moreover, we say a function f has finite energy if E( f ) < ∞.

2.3 Intrinsic metrics

Next, we introduce the concept of intrinsic metrics. A pseudo metric is a symmetric mapX × X → [0,∞) with zero diagonal which satisfies the triangle inequality.

Definition 2.2 (Intrinsic metric) A pseudo metric ρ on X is called an intrinsic metric if∑y∈X

μxyρ2(x, y) ≤ m(x), x ∈ X.

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If for a function f : X → R the map ( f ) : x → ∑y∈X μxy |∇xy f |2 takes finite

values, then ( f ) defines the energy measure of f . Thus, a pseudo metric ρ is intrinsic ifthe energy measures (ρ(x, ·)), x ∈ X , are absolutely continuous with respect to m withRadon-Nikodym derivative d

dm (ρ(x, ·)) = (ρ(x, ·))/m satisfying (ρ(x, ·))/m ≤ 1.In various situations the natural graph distance proves to be insufficient for the investiga-

tions of unbounded Laplacians, see [36,56,57]. For this reason the concept of intrinsic metricsdeveloped in [12] for regular Dirichlet forms received quite some attention as a candidate toovercome these problems. Indeed, intrinsic metrics already have been applied successfullyto various problems on graphs [3,4,13,14,24] and related settings [15].

The jumps size s of a pseudo metric is given by

s := supρ(x, y) | x, y ∈ X, x ∼ y ∈ [0,∞].From now on, ρ always denotes an intrinsic metric and s denotes its jump size.We fix a base point o ∈ X which we suppress in notation and denote the distance balls by

Br = x ∈ X | ρ(x, o) ≤ r, r ≥ 0.

Since ρ takes values in [0,∞) in our setting, the results are indeed independent of the choiceof o. For U ⊆ X , we write Br (U ) = x ∈ X | ρ(x, y) ≤ r for some y ∈ U , r ≥ 0.

Define the weighted vertex degree Deg : X → [0,∞) by

Deg(x) = 1

m(x)

∑y∈X

μxy, x ∈ X.

Definition 2.3 (Compatible metric) A pseudo metric on X is called compatible if it has finitejump size and the restriction of Deg to every distance ball is bounded, i.e., Deg|Br ≤ C(r) <

∞ for all r ≥ 0.

Example 2.4 (a) For any given weighted graph there is an intrinsic path metric defined by

δ(x, y) = infx=x0∼···∼xn=y

n−1∑i=0

(Deg(xi ) ∨ Deg(xi+1))− 1

2 .

This intrinsic metric can be turned into an intrinsic metric δr with finite jump size s = rby taking the path metric with edge weights δ(x, y) ∧ r , x ∼ y. In many cases, neitherδr nor δ is compatible.

(b) If the measure m is larger than the measure n(x) = ∑y∈X μxy , x ∈ X , then the natural

graph distance (i.e., the path metric with edge weights 1) is an intrinsic metric which iscompatible since s = 1 and Deg ≤ 1 in this case.

Remark 2.5 (a) In view of Example 2.4 (b) it is apparent that [20, Theorem 1.1] is includedin Theorem 1.1.

(b) In [24, Theorem A.1] a Hopf-Rinow type theorem is shown which states that for a locallyfinite graph a path metric is complete if and only if all balls are finite. Thus, compatibilitycan be seen as a completeness assumption of the graph.

(c) It is not hard to see that there are graphs that do not allow for a compatible intrinsic metric.However, to a given edge weight function μ and a pseudo metric ρ, we can always assigna minimal measure m such that ρ is intrinsic, i.e., let m(x) = ∑

y∈X μxyρ2(x, y), x ∈ X .

If ρ already has finite jump size and all balls are finite, then ρ is automatically compatible.(d) The assumption that Deg is bounded on distance balls is equivalent to either of the

following assumptions

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(i) The restriction of to any distance ball (with Dirichlet boundary conditions) is abounded operator.

(ii) The Radon-Nikodym derivative of the measure n given by n(x) = ∑y μxy , x ∈ X ,

with respect to the measure m is bounded on the distance balls .

The equivalence of (i) follows from Theorem [23, Theorem 9.3] and the one of (ii) isobvious.

In the subsequent, we will make use of the cut-off function η = ηr,R , 0 ≤ r < R, on Xgiven by

η = 1 ∧(

R − ρ(·, o)

R − r

)+

.

Lemma 2.6 Let η = ηr,R, 0 < r < R, be given as above. Then,

(a) η|Br ≡ 1 and η|X\BR ≡ 0.(b) For x ∈ X,

∑y∈X

μxy |∇xyη|2 ≤ 1

(R − r)2 1BR+s\Br−s (x)m(x).

Proof (a) is obvious from the definition of η and (b) follows directly from |∇xyη| ≤1

R−r ρ(x, y)1BR+s\Br−s (x) for x ∼ y and the intrinsic metric property of ρ. 2.4 Green’s formula, Leibniz rules and mean value theorem

We first prove a Green’s formula which is an L p version of the one in [24].

Lemma 2.7 (Green’s formula) Let p ∈ [1,∞), U ⊆ X and assume Deg is bounded on U.

Then for all f with f 1U ∈ L p(X, m) ∩ F(X) and g ∈ Lp∗

(X, m) with Bs(supp g) ⊆ U

∑x∈X

( f )(x)g(x)m(x) = 1

2

∑x,y∈U

μxy∇xy f ∇xy g.

Proof The formal calculation in the proof of Green’s formula is a straightforward algebraicmanipulation. To ensure that all involved terms converge absolutely, one invokes Hölder’sinequality and the boundedness assumption on Deg (confer the proof of Lemma 3.1 and 3.3in [24]).

The following Leibniz rules follow by direct computations.

Lemma 2.8 (Leibniz rules) For all x, y ∈ X, x ∼ y and f, g : X → R

∇xy( f g) = f (y)∇xy g + g(x)∇xy f

= f (y)∇xy g + g(y)∇xy f + ∇xy f ∇xy g.

A fundamental difference of Laplacians on graphs and on manifolds is the absence ofa chain rule in the graph case. In particular, existence of a chain rule can be used as acharacterization for a regular Dirichlet form to be strongly local. We circumvent this problemby using the mean value theorem from calculus instead. In particular, for a continuouslydifferentiable function φ : R → R and f : X → R, we have

∇xy(φ f ) = φ′(ζ )∇xy f, for some ζ ∈ [ f (x) ∧ f (y), f (x) ∨ f (y)].

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In this paper we will apply this formula to get estimates for the function φ : t → t p−1,p ∈ (1,∞). However, we need a refined inequality as it was already used in the proof of [25,Theorem 2.1]. For the convenience of the reader, we include a short proof here.

Lemma 2.9 (Mean value inequalities) For all f : X → R and x ∼ y with ∇xy f ≥ 0,

(a) ∇xy f p−1 ≥ 12 ( f p−2(x) + f p−2(y))∇xy f , for p ∈ [2,∞),

(b) ∇xy f p−1 ≥ C( f (x) ∨ f (y))p−2∇xy f , for p ∈ (1,∞), where C = (p − 1) ∧ 1.

Proof (a) Denote a = f (y), b = f (x). As it is the only non-trivial case, we assume0 < a < b. Note that for p = 1

bp−1 − a p−1 = (b − a)(bp−2 + a p−2) + ab(bp−3 − a p−3).

Thus, the statement is immediate for p ≥ 3 since the second term on the right side is non-negative in this case. Let 2 ≤ p < 3 and note a p−3 > bp−3. The function t → t2−p isconvex on (0,∞) and, thus, its image lies below the line segment connecting (b−1, bp−2)

and (a−1, a p−2). Therefore,

a p−3−bp−3 ≤ a p−3−bp−3

(3 − p)=

a−1∫

b−1

t2−pdt ≤ (a−1 − b−1)

((bp−2 − a p−2)

2+a p−2

)

= 1

2ab(b − a)(a p−2 + bp−2).

From the equality in the beginning of the proof we now deduce the assertion in the case2 ≤ p < 3.

(b) The case p ≥ 2 follows from (a). The case 1 < p ≤ 2 in (b) follows directly from themean value theorem.

3 Proofs for harmonic functions

In this section we prove the main theorems and the corresponding corollaries for harmonicfunctions. It will be convenient to introduce the following orientation on the edges. For agiven non-negative subharmonic function f , we let E f be the set of oriented edges e = e+e−such that

∇e f ≥ 0, i.e., f (e+) ≥ f (e−).

3.1 The key estimate

The lemma below is vital for the proof of Theorems 1.1 and 1.8.

Lemma 3.1 Let p ∈ (1,∞), 0 ≤ ϕ ∈ L∞(X) and U = Bs(supp ϕ). Assume Deg is boundedon U. Then, for every non-negative subharmonic function f with f 1U ∈ L p(X, m),∑

e∈E f

μe f p−2(e+)ϕ2(e−)|∇e f |2 ≤ C∑

e∈E f ,e⊂U μe f p−1(e+)ϕ(e−)∇e f |∇eϕ|,

where C = 2/((p − 1) ∧ 1).

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Proof From the assumptions f 1U ∈ L p(X, m) and ϕ ∈ L∞(X), we infer ϕ2 f p−1 ∈L p∗

(X, m) (as p∗ = p/(p − 1)). Thus, compatibility of the pseudo metric implies applica-bility of Green’s formula with f and g = ϕ2 f p−1. We start by using non-negativity andsubharmonicity of f before applying Green’s formula (Lemma 2.7) and the first and secondLeibniz rule (Lemma 2.8)

0 ≥∑x∈X

( f )(x)(ϕ2 f p−1)(x)m(x) =∑

e∈E f ,e⊂U

μe∇e f ∇e(ϕ2 f p−1)

=∑e⊂U

μe∇e f[ϕ2(e−)∇e f p−1 + f p−1(e+)∇eϕ

2]

=∑e⊂U

μe∇e f[ϕ2(e−)∇e f p−1 + 2 f p−1(e+)ϕ(e−)∇eϕ + f p−1(e+)|∇eϕ|2]

≥ C∑e⊂U

μe f p−2(e+)ϕ2(e−)|∇e f |2 + 2∑e⊂U

μe f p−1(e+)ϕ(e−)∇e f ∇eϕ,

where we dropped the third term in the third line since it is non-negative because of ∇e f ≥0 and we estimated the first term on the right hand side using the mean value theorem,Lemma 2.9 (b). Absolute convergence of the two terms in the last line can be checked usingHölder’s inequality and the assumptions f 1U ∈ L p(X, m), ϕ ∈ L∞(X) and boundednessof Deg on U . Hence, we obtain the statement of the lemma. 3.2 Proof of Karp’s theorem

Proof of Theorem 1.1 Let p ∈ (1,∞) and let f be a non-negative subharmonic function.Assume f 1Br ∈ L p(X, m) for all r ≥ 0 since otherwise infr0

∫ ∞r0

r/‖ f 1Br ‖ppdr = 0. Let

η = ηr+s,R−s with 0 < r < R −3s (see Sect. 2.3). Then by Lemma 3.1 (applied with ϕ = η)we obtain (noting additionally that ∇xyη = 0, x, y ∈ Br )

∑e⊂BR

μe f p−2(e+)η2(e−)|∇e f |2 ≤ C∑

e⊂BR\Brμe f p−1(e+)η(e−)∇e f |∇eη|.

Now, the Cauchy-Schwarz inequality,∑

e μe f p(e+)|∇eη|2 ≤ ∑x,y μxy f p(x)|∇xyη|2 and

the cut-off function lemma, Lemma 2.6, yield

⎛⎝ ∑

e⊂BR

μe f p−2(e+)η2(e−)|∇e f |2⎞⎠

2

≤ C

⎛⎝ ∑

e⊂BR\Br

μe f p(e+)|∇eη|2⎞⎠

⎛⎝ ∑

e⊂BR\Br

f p−2(e+)η2(e−)|∇e f |2⎞⎠

≤ C

(R − r)2 ‖ f 1BR\Br ‖pp

⎛⎝

⎛⎝ ∑

e⊂BR

−∑

e⊂Br

⎞⎠ f p−2(e+)η2(e−)|∇e f |2

⎞⎠ .

Let R0 ≥ 3s be such that f 1BR0= 0 and denote

v(r) = ‖ f 1Br ‖pp, r ≥ 0.

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Moreover, for j ≥ 0, let R j = 2 j R0, ϕ j = ηR j +s,R j+1−s and

Q j+1 = ∑e⊂BR j+1

μe f p−2(e+)ϕ2j (e−)|∇e f |2.

As ϕ j−1 ≤ ϕ j , we get Q j ≤ Q j+1 and together with the estimate above this implies

Q j Q j+1 ≤ Q2j+1 ≤ C

v(R j+1)

(R j+1 − R j )2 (Q j+1 − Q j ), j ≥ 0.

Since R j+1 = 2R j , dividing the above inequality byv(R j+1)

R2j+1

Q j Q j+1 and adding C/Q j+1

yield

R2j+1

v(R j+1)+ C

Q j+1≤ C

Q j

and, thus,

1

C

∞∑j=1

R2j+1

v(R j+1)≤ 1

Q1.

Now, the assumption∫ ∞

R0r/v(r)dr = ∞ implies

∑∞j=0

R2j

v(R j )= ∞. Therefore, Q1 = 0. As

this is true for all R0 large enough, we have

f p−2(e+)|∇e f |2 = 0,

for all edges e. For p ≥ 2, connectedness clearly implies that f is constant. On the otherhand, for p ∈ (1, 2], we always have f p−2(e+) > 0 and, thus, f is constant. 3.3 Proof of the Caccioppoli inequality

Proof of Theorem 1.8 Using Lemma 3.1 and the inequality ab ≤ εa2 + 14ε

b2, ε > 0, weestimate

∑e∈E f

μe f p−2(e+)ϕ2(e−)|∇e f |2 ≤ C∑

e∈E f

μe f p−1(e+)ϕ(e−)∇e f |∇eϕ|

≤ 1

2

∑e∈E f

μe f p−2(e+)ϕ2(e−)|∇e f |2 + C∑

e∈E f

μe f p(e+)|∇eϕ|2.

Letting ϕ = η = ηr+s,R−s with 0 < r < R − 3s (from Sect. 2.3) and using the cut-offfunction lemma, Lemma 2.6, we arrive at

∑e∈E f

μe f p−2(e+)|∇e f |2 ≤ C∑

e∈E fμe f p(e+)|∇eη|2 ≤ C

(R−r)2 ‖ f 1BR\Br ‖pp.

Remark 3.2 In order to obtain the stronger statement for p ∈ [2,∞) mentioned inRemark 1.9 (b), we invoke Lemma 2.9 (a) in the proof of Lemma 3.1 instead of Lemma 2.9 (b)and proceed as in the proof above.

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3.4 Proof of the corollaries

In this section we prove the corollaries.

Proof of Corollary 1.2 (Yau’s L p Liouville theorem) Clearly the integral in Theorem 1.1diverges if f ∈ L p(X, m). Proof of Corollary 1.4 (Domain of the L p generators) Let f ∈ L p(X, m) ∩ F(X) be suchthat ( + 1) f = 0. Since the positive and negative part f+, f− of f are non-negative,subharmonic and in L p(X, m), they must be constant by Corollary 1.2. This implies f± ≡ 0and, thus, f ≡ 0. Now, the proof of the corollary works literally line by line as the proof of[35, Theorem 5].

For the proof of Corollary 1.6 we recall the following well known equivalent conditionsfor recurrence.

Proposition 3.3 (Characterization of recurrence) Let a connected graph X be given. Thenthe following are equivalent.

(i) For the transition matrix P with Px,y = μxy/∑

z∈X μxz , x, y ∈ X, we have∑∞n=0 P(n)(x, y) = ∞ for some (all) x, y ∈ X, where P(n) denotes the nth power

of P.(ii) For m ≡ 1 and some (all) x, y ∈ X, we have

∫ ∞0 e−t Lδx (y)dt = ∞, where δx (y) = 1

if x = y and zero otherwise.(iii) For all m and some (all) x, y ∈ X, we have

∫ ∞0 e−t Lδx (y)dt = ∞.

(iv) Every bounded superharmonic (or subharmonic) function is constant.(v) Every non-negative superharmonic function is constant.

(vi) Every superharmonic (or subharmonic) function of finite energy is constant.(vii) cap(x) := infE( f )| f ∈ Cc(X), f (x) = 1 = 0 for some (all) x ∈ X

A graph is called recurrent if one of the equivalent statements of Proposition 3.3 is satisfied.

Proof The equivalence (i)⇔(ii) is shown in [47, Theorem 6] (confer [7, Theorem 4.34]).The equivalences (ii)⇔(vi)⇔(iii) are in [47, Theorem 2 and Theorem 9] (confer [48, Theo-rem 3.34]). The equivalences (i)⇔(v)⇔(vii) are found in [55, Theorem 1.16, Theorem 2.12].The equivalence (iv)⇔(v) follows since every non-negative superharmonic function f canbe approximated by the bounded superharmonic functions f ∧ n, n ≥ 1. Proof of Corollary 1.6 (Recurrence) Theorem 1.1 implies that any non-negative boundedsubharmonic function f is constant provided infr0

∫ ∞r0

r/m(Br )dr = ∞ since ‖ f 1Br ‖pp ≤

‖ f ‖p∞m(Br ), r ≥ 0. By Proposition 3.3 the graph is recurrent.

4 L1-Liouville theorem and counter-examples

In this section we deal with the borderline case of the L p Liouville theorem p = 1. We firstprove Theorem 1.7 which deals with the stochastic complete case and then give two exampleswhich show that there is no L1 Liouville theorem for non-negative subharmonic functionsin the general case.

A graph is called stochastically complete if e−t L 1 = 1, where 1 denotes the functionthat is constantly one on X . For the relevance of the concept see [17,35,56]. The proof ofTheorem 1.7 follows along the lines of the proof of [17, Theorem 13.2].

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Proof of Theorem 1.7 If the graph is recurrent, then there are no non-constant non-negativesuperharmonic functions by Proposition 3.3. So assume the graph is not recurrent whichimplies G(x, y) = ∫ ∞

0 e−t Lδx (y)dt < ∞, x, y ∈ X , again by Proposition 3.3. Let Kn ,n ≥ 0, be an sequence of finite sets exhausting X and Gn(x, y) = ∫ ∞

0 e−t Ln δx (y)dt ,where Ln are the finite dimensional operators arising from the restriction of the form Qto Cc(Kn). By domain monotonicity, [35, Proposition 2.6 and 2.7] the semigroups e−t Ln

converge monotonously increasing to e−t L and, hence, Gn(x, y) ≤ G(x, y) for x, y ∈ Kn ,and Gn G, n → ∞, pointwise. By direct calculation for any x ∈ Kn

LnGn(x, y) =∞∫

0

Lne−t Ln δx (y)dt =∞∫

0

∂t e−t Ln δx (y)dt = [e−t Ln δx (y)]∞0 = δx (y)

and, hence, Gn(x, ·) are harmonic on Kn \ x, n ≥ 0.Let u be a non-trivial non-negative superharmonic function which is strictly positive by

the minimum principle [35, Theorem 8].Let U ⊆ X be finite with o ∈ U ⊆ Kn , n ≥ 0and C > 0 be such that Cu ≥ G(o, ·) on U . By the minimum principle Cu ≥ Gn(o, ·) onKn \o and, hence, Cu ≥ G(o, ·) on X by the discussion above. If the graph is stochasticallycomplete, then we get by Fubini’s theorem

C‖u‖1 ≥ ‖G(o, ·)‖1 =∞∫

0

∑x∈X

e−t Lδo(x)m(x)dt =∞∫

0

e−t L 1(o)dt =∞∫

0

dt = ∞.

Hence, u is not in L1(X, m). In the proof we show that in the non-recurrent case there are no nontrivial superharmonic

functions in L1. This is explained since in the case of finite measure recurrence and stochasticcompleteness are equivalent [47, Theorem 12].

Next, we show that in general there is no L p Liouville theorem for p ∈ (0, 1]. This isanalogous to the situation in Riemannian geometry, where counter-examples were given by[8,41]. Our first example is a graph of finite volume and the second is of infinite volume.

Example 4.1 (Finite volume) Let G = (X, μ, m) be an infinite line graph, i.e., X = Z andx ∼ y iff |x−y| = 1 for x, y ∈ Z. Define the edge weight by μxy = 21−(|x |∨|y|) for x ∼ y andthe measure m by m(x) = (|x | + 1)−22−|x |, x ∈ Z, which implies m(X) < ∞. The intrinsicmetric δ (introduced in Example 2.4) is compatible as it satisfies δ(x, x + 1) ≥ C(|x |+ 1)−1

and, thus,∑∞

x=−∞ δ(x, x + 1) = ∞. However, the function f defined as

f (x) = sign(x)(2|x | − 1), x ∈ Z,

is harmonic and, clearly, f ∈ L p(X, m), p ∈ (0, 1].Example 4.2 (Infinite volume) We can extend the example above to the infinite volume case.Let G be the graph from above and G ′ be a locally finite graph of infinite volume whichallows for a compatible path metric. We glue G ′ to the vertex x = 0 of the graph G byidentifying a vertex in G ′ with x = 0. Next, we extend the path metrics in the natural wayand obtain (by renormalizing the edge weights of the metric at the edges around x = 0 ifnecessary) again a compatible intrinsic metric and the graph has infinite volume. Moreover,we extend f on G from above by zero to G ′ and obtain a harmonic function which is in L p ,p ∈ (0, 1].

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5 Applications of Karp’s theorem

In this section we prove Theorem 1.10 and give several applications which mainly circlearound the case of finite measure.

Proof of Theorem 1.10 We assume that for some p ∈ (1,∞) the non-negative subharmonicfunction f is in L p(X, mρ−2

1 ) and, hence, f pρ−21 ∈ L1(X, m). For large r0 ≥ 1, we estimate

∞∫

r0

r

‖ f 1Br ‖pp

dr ≥∞∫

r0

r

r2‖ f pρ−21 1Br ‖1

dr ≥ C

∞∫

r0

1

rdr = ∞.

Hence, Theorem 1.1 implies that f is constant.

Next, we turn to several consequences of Theorem 1.10. A function f : X → R is said togrow less than a function g : [0,∞) → (0,∞) if there are β ∈ (0, 1) and C > 0 such that

f (x) ≤ Cgβ(ρ1(x)), x ∈ X.

We say f grows polynomially if f grows less than a polynomial.We say the measure m has a finite q th moment, q ∈ R, with respect to an intrinsic metric

ρ if

ρ1 ∈ Lq(X, m),

where ρ1 = 1 ∨ ρ(·, o). This assumption implies that all balls have finite measure and ifq ≥ 0 it also implies m(X) < ∞.

Corollary 5.1 (Measures with finite moments) Assume a connected weighted graph allowsfor a compatible intrinsic metric and the measure has a finite qth moment, q ∈ R. Thenevery non-negative subharmonic function f that grows less than r → rq+2 is constant. Inparticular, if q > −2, then boundedness of f implies f is constant.

Proof If f grows less than r → rq+2, then there is ε > 0 such that f 1+ερ−21 ≤ Cρ

q1 on X .

By the assumption ρ1 ∈ Lq(X, m) it follows f ∈ L p(X, mρ−21 ) for p = 1 + ε. Hence, the

assertion follows from Theorem 1.10.

Letting q = 0 in the above theorem gives the following immediate corollary.

Corollary 5.2 (Finite measure) Assume a connected weighted graph X allows for a compat-ible intrinsic metric and m(X) < ∞. Then every non-negative subharmonic function f thatgrows less than quadratic is constant. In particular, f ∈ L∞(X) implies that f is constant.

The final corollary of this section is a consequence of Corollary 1.2.

Corollary 5.3 (Exponentially decaying measure) Assume a connected weighted graph Xallows for a compatible intrinsic metric and m(X) < ∞, and there is β > 0 such that

lim supr→∞

1

rβlog m(Br+1 \ Br ) < 0.

Then every non-negative subharmonic function that grows polynomially is constant.

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Proof If a non-negative subharmonic function f grows polynomially, then there is q > 0such that

‖ f ‖pp ≤ C

∑x∈X

ρq1 (x, o)m(x) ≤ C

∞∑r=1

rqm(Br \ Br−1) + C < ∞

by the assumption on the measure. Hence, the theorem follows from Corollary 1.2.

6 Applications to harmonic maps

Harmonic maps between metric measure spaces were introduced by Jost [27–30] and har-monic maps from graphs into Riemannian manifolds or metric spaces have been studied bymany authors, e.g. [26,31,39] and and for alternative definitions, see [19,37,38,50,52].

We use our results concerning the function theory on graphs to derive various Liouvilletype theorems for harmonic maps from graphs. A particular focus lies on bounded harmonicmaps and harmonic maps of finite energy.

Let (X, μ, m) be a weighted graph. We briefly recall the set up of Hadamard spaces andharmonic maps.

A complete geodesic space (Y, d) is called an NPC space if it locally satisfies Toponogov’striangle comparison for non-positive sectional curvature. We refer to Burago–Burago–Ivanov[1], Jost [29] and Bridson–Haefliger[2] for definitions. Here NPC stands for “non-positivecurvature” in the sense of Alexandrov. The space (Y, d) is called an Hadamard space, ifthe Toponogov’s triangle comparison holds globally, i.e., holds for arbitrary large geodesictriangles. A simply connected NPC space is an Hadamard space, see [1]. For the sake ofsimplicity, we only consider Hadamard spaces, also called CAT(0) spaces, as targets ofharmonic maps X → Y . For general NPC spaces, we may pass to the universal covers of Xand Y, and consider the equivariant harmonic maps, see Jost [27,28].

Let b : P1(Y ) → Y denote the barycenter map on Y , where P1(Y ) is the space ofprobability measures on Y with finite first moment, that is, b(ν) is the barycenter of theprobability measure ν on Y, see e.g. Sturm [51, Propositon 4.3] and confer [40, Definition 2.2and Example 1].

We define the random walk measure Px of x ∈ X by

Px (y) := μxy∑z∈X μxz

and denote by u∗ Px the push forward of the probability measure Px under the map u :X → Y . In order to carry out the barycenter construction for a map u pointwise, we needu∗ Px ∈ P1(Y ) which means that u∗ Px has finite first moment. Thus, similar to the harmonicfunction case, we define a class of maps

F(X, Y ) :=⎧⎨⎩u : X → Y |

∑y∈X

d(u(y), y0)Px (y) < ∞ for all x ∈ X, y0 ∈ Y

⎫⎬⎭ .

Definition 6.1 (Harmonic map) A map u : X → Y is called a harmonic map if u ∈ F(X, Y )

and for every x ∈ X

u(x) = b(u∗ Px ).

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One immediately finds that the measure m plays no role in the definition of harmonic maps.In the following, we always denote by u : X → Y a harmonic map from a weighted graphinto an Hadamard space.

6.1 Proof of Theorem 1.11

The proof of Theorem 1.11 is a rather immediate consequence of Theorem 1.10 and thefollowing lemma which is a consequence of Jensen’s inequality and convexity of distancefunctions on Hadamard spaces.

Lemma 6.2 For every harmonic map u : X → Y where Y is an Hadamard space, thefunctions X → [0,∞), x → d(u(x), y), for fixed y ∈ Y , are subharmonic.

Proof Jensen’s inequality in Hadamard spaces, see [51, Theorem 6.2], states that for everylower semi-continuous convex function g : Y → [0,∞) and ν ∈ P1(Y )

g(b(ν)) ≤∫

Y

g(y)ν(dy).

Now any distance function y → d(y, y0) to a point y0 ∈ Y is convex in an Hadamard space,see e.g. [1, Corollary 9.2.14], which yields the statement. Proof of Theorem 1.11 Combining Theorem 1.10 and Lemma 6.2 yields that x →d(u(x), y) is constant. Hence, u is bounded. If mρ−2

1 (X) = ∑x∈X m(x)ρ−2

1 (x) = ∞,then a constant function is in L p(X, mρ−2

1 ) if and only if it is zero. Moreover, if y is in theimage of u then d(u(·), y) ≡ 0 and hence u(x) = y for all x ∈ X . 6.2 Harmonic maps of finite energy

In this section we consider harmonic maps of finite energy and prove Theorems 1.12 and1.13 which are analogues to theorems of Cheng–Tam–Wang [9] from Riemannian geometry.We say a harmonic map u : X → Y has finite energy if

1

2

∑x,y∈X

μxyd2(u(x), u(y)) < ∞.

In order to do so, we need the equivalence of boundedness of finite energy harmonicfunctions on a graph and that of non-negative subharmonic functions. Recall that a functionf : X → R is said to have finite energy if E( f ) < ∞, see Sect. 2.2. In Riemannian geometrysuch a theorem was first proven in [9, Theorem 1.2]. We give a different proof here in thediscrete setting by Royden’s decomposition.

Theorem 6.3 For connected weighted graphs every harmonic function with finite energy isbounded if and only if every non-negative subharmonic function with finite energy is bounded.

Proof As positive and negative part of a harmonic function are non-negative and subhar-monic functions, boundedness of non-negative subharmonic functions of finite energy impliesboundedness of harmonic functions of finite energy. We now turn to the other direction. ByProposition 3.3 there are no non-constant subharmonic functions of finite energy in the casethe graph is recurrent. Therefore, we assume the graph is not recurrent (also called transientin the connected case). Let f be a non-negative subharmonic function with finite energy.Then by the discrete version of Royden’s decomposition theorem, see [48, Theorem 3.69],

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there are unique functions g and h where g is in the completion of Cc(X) under the norm‖ϕ‖o = (E(ϕ) + ϕ(o)2)1/2, ϕ ∈ Cc(X), and h is a harmonic function of finite energy suchthat

f = g + h and E( f ) = E(g) + E(h)

By [48, Lemma 3.70], g ≤ 0 since g is subharmonic. Therefore, 0 ≤ f ≤ h. By assumptionh is bounded and, therefore, f is bounded. Proof of Theorem 1.12 Let u : X → Y be a harmonic map of finite energy. For some fixedy0 ∈ Y the function f = d(u(·), y0) is non-negative and subharmonic by Lemma 6.2.Furthermore, by the triangle inequality and the assumption that u has finite energy we get

E( f ) = 1

2

∑x,y

μxy( f (x) − f (y))2 ≤ 1

2

∑x,y∈X

μxyd2(u(x), u(y)) < ∞.

Now, by Theorem 6.3 we get that f as a non-negative subharmonic function of finite energymust be bounded whenever every harmonic function of finite energy on X is bounded (whichis our assumption). Thus, u is bounded.

Theorem 1.13 is a consequence of Theorem 1.12 and the theorem of Kuwae–Sturm [40]below which goes back to Kendall in the manifold case [33, Theorem 6] (confer [22,40,42]).However, although it is not explicitly mentioned in [40] one actually needs an additionalassumption on the local compactness of the target, i.e., every point has a precompact neigh-borhood.

Theorem 6.4 (Kendall’s theorem [40, Theorem 3.1]) Assume that on a connected weightedgraph every bounded harmonic functions is constant. Then, every bounded harmonic mapinto a locally compact Hadamard space is constant.

Next, we come to the proof of Theorem 1.13.

Proof of Theorem 1.13 Let f be a harmonic function on X of finite energy. By a discreteversion of Virtanen’s theorem, see [48, Theorem 3.73], f can be approximated by boundedharmonic functions fn of finite energy (with respect to the norm ‖ϕ‖o = (E(ϕ)+ϕ(o)2)1/2).By assumption the functions fn , n ≥ 1, are constant and, thus, f must be constant. Theo-rem 1.12 implies now that any harmonic map is bounded and, thus, Theorem 6.4 implies thatevery harmonic map is constant. 6.3 Harmonic maps and assumptions on the measure of X

In this subsection we collect several quantitative results that follow from what we have provenbefore.

The first corollary can be seen as an analogue to Yau’s L p-Liouville type theorem.

Corollary 6.5 Assume a connected weighted graph X allows for a compatible intrinsicmetric and let u be a harmonic map into an Hadamard space Y . If there is y ∈ Y such thatd(u(·), y) ∈ L p(X, m) for some p ∈ (1,∞), then u is bounded. If additionally m(X) = ∞,then u is constant.

Proof The function d(u(·), y) is subharmonic, by Lemma 6.2, and in L p(X, m) by assump-tion. Hence, Corollary 1.2 yields d(u(·), y) is constant. The assumption of infinite measureimplies d(u(x), y) = 0 for all x ∈ X.

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We say a harmonic map u into an Hadamard space (Y, d) grows less than a functiong : [0,∞) → (0,∞) if d(u(·), y) grows less than g for some y ∈ Y (confer Sect. 5). Thenext two corollaries are analogues of Corollaries 5.1 and 5.2.

Corollary 6.6 (Measures with finite moment—harmonic maps) Assume a connectedweighted graph allows for a compatible intrinsic metric and the measure has a finite qthmoment, q > −2. Then every harmonic map into an Hadamard space that grows less thatr → rq+2 is constant. In particular, bounded harmonic maps and harmonic maps with finiteenergy are constant.

Proof Let u be a harmonic map. Since we assume q + 2 > 0, we get by the triangleinequality that for all y ∈ Y the subharmonic (Lemma 6.2) functions d(u(·), y) grow lessthan r → rq+2. Hence, by Corollary 5.1 the subharmonic function d(u(·), y) is constant forall y which implies that u is constant. This proves the first assertion. Since q + 2 > 0, itis easy to see that every bounded harmonic function on X is constant. The second assertionfollows from Theorems 6.4 and 1.13. Corollary 6.7 (Finite measure—harmonic maps) Assume a connected weighted graph Xallows for a compatible intrinsic metric and m(X) < ∞. Then every harmonic map into anHadamard space that grows less than quadratic is constant. In particular, bounded harmonicmaps and harmonic maps with finite energy are constant.

Proof The statements follow directly by the corollary above putting q = 0. Finally we say that a harmonic map grows polynomially if it grows less than a polynomial

and state a corollary analogous to Corollary 5.3.

Theorem 6.8 (Exponentially decaying measure—harmonic maps) Assume a connectedweighted graph X allows for a compatible intrinsic metric, m(X) < ∞ and there is β > 0such that

lim supr→∞

1

rβlog m(Br+1 \ Br ) < 0.

Then every harmonic map into an Hadamard space that grows polynomially is constant. Inparticular, bounded harmonic maps and harmonic maps with finite energy are constant.

Proof The statements follow from Corollary 5.3, Theorems 6.4 and 1.13. Acknowledgments BH thanks Jürgen Jost for inspiring discussions on L p Liouville theorem and constantsupport, and acknowledges the financial support from the funding of the European Research Council underthe European Union’s Seventh Framework Programme (FP7/2007-2013) / ERC grant agreement no 267087.MK enjoyed discussions with Gabor Lippner, Dan Mangoubi, Marcel Schmidt and Radosław Wojciechowskion the subject and acknowledges the financial support of the German Science Foundation (DFG), Golda MeirFellowship, the Israel Science Foundation (grant no. 1105/10 and no. 225/10) and BSF grant no. 2010214.

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X. Huang, M. Keller, J. Masamune, R.Wojciechowski, A note on self-adjoint extensionsof the Laplacian on weighted graphs, Journal of

Functional Analysis 265 (2013), 1556–1578.

215

Available online at www.sciencedirect.com

Journal of Functional Analysis 265 (2013) 1556–1578

www.elsevier.com/locate/jfa

A note on self-adjoint extensions of the Laplacian onweighted graphs

Xueping Huang a, Matthias Keller b, Jun Masamune c, RadosławK. Wojciechowski d,∗

a Fakultät für Mathematik, Universität Bielefeld, 33501 Bielefeld, Germanyb Mathematisches Institut, Friedrich-Schiller-Universität Jena, 07743 Jena, Germany

c Research Center for Pure and Applied Mathematics, Graduate School of Information Sciences, Tohoku University,6-3-09 Aramaki-Aza-Aoba, Aoba-ku, Sendai 980-8579, Japan

d Department of Mathematics and Computer Science, York College of the City University of New York,94-20 Guy R. Brewer Blvd., Jamaica, NY 11451, USA

Received 31 August 2012; accepted 3 June 2013

Available online 5 July 2013

Communicated by Daniel W. Stroock

Abstract

We study the uniqueness of self-adjoint and Markovian extensions of the Laplacian on weighted graphs.We first show that, for locally finite graphs and a certain family of metrics, completeness of the graphimplies uniqueness of these extensions. Moreover, in the case when the graph is not metrically completeand the Cauchy boundary has finite capacity, we characterize the uniqueness of the Markovian extensions.© 2013 Elsevier Inc. All rights reserved.

Keywords: Weighted graphs; Laplacians; Essential self-adjointness; Intrinsic metrics

1. Introduction

Determining the uniqueness of self-adjoint extensions of a symmetric operator in a certainclass is a fundamental topic of functional analysis going back to the work of Friedrichs and vonNeumann [12,45]. If an operator has a unique self-adjoint extension, then it is called essentially

* Corresponding author.E-mail addresses: [email protected] (X. Huang), [email protected] (M. Keller),

[email protected] (J. Masamune), [email protected] (R.K. Wojciechowski).

0022-1236/$ – see front matter © 2013 Elsevier Inc. All rights reserved.http://dx.doi.org/10.1016/j.jfa.2013.06.004

X. Huang et al. / Journal of Functional Analysis 265 (2013) 1556–1578 1557

self-adjoint. A self-adjoint extension whose corresponding form is a Dirichlet form is calledMarkovian and, when such an extension is unique, the operator is said to have a unique Marko-vian extension. It is clear that essential self-adjointness implies the uniqueness of Markovianextensions, but the converse is not necessarily true as can be seen by examples.

In the case of Riemannian manifolds, the (minimal) Laplacian, whose domain is the space ofsmooth functions with compact support, has Markovian extensions and generates the Brownianmotion. (The Laplacian should satisfy, in addition, the regularity property, but there is alwaysan equivalent operator which has this property [13].) A well-known result going back to thework of Gaffney [15,16] essentially states that, on a geodesically complete manifold, the Lapla-cian has a unique Markovian extension. (In [15,16] the so-called Gaffney Laplacian was provento be essentially self-adjoint instead of the minimal one. The essential self-adjointness of theGaffney Laplacian is equivalent to the uniqueness of Markovian extensions of the minimal Lapla-cian. Indeed, Gaffney’s result states that the condition W

1,20 = W 1,2, which is equivalent to the

uniqueness of Markovian extensions of the minimal Laplacian [22], implies the essential self-adjointness of the Gaffney Laplacian. The converse implication was proven in [38].) Later, itwas shown that the Laplacian on a geodesically complete Riemannian manifold is essentiallyself-adjoint [2,43]. On the other hand, if the manifold is geodesically incomplete, the Laplacianis not essentially self-adjoint in general; however, if the Cauchy boundary, which is the differ-ence between the completion of the manifold and the manifold itself, is “small” in some sense,then the Laplacian is essentially self-adjoint or has a unique Markov extension depending onhow small the Cauchy boundary is [1,3,22,35–37] (see also the references therein). For stronglylocal regular Dirichlet forms, the uniqueness of Silverstein extensions was proven by Kawabataand Takeda [30] in the case when the underlying space is metrically complete with respect tothe Carnot–Caratheodori distance. This result was extended to general regular Dirichlet formsby Kuwae and Shiozawa [34] using the intrinsic distance defined by Frank, Lenz, and Wingertin [11].

Recently, there has been a tremendous amount of work devoted to the study of self-adjointextensions of certain operators defined on graphs. More specifically, these issues are studied foradjacency, (magnetic) Laplacian, and Schrödinger-type operators on graphs in [4–6,17–19,24,28,29,31,32,35,39,41,42,44,46–48] among others.

Let us mention, in particular, the series of papers [4,5,44] by Colin de Verdière, Torki-Hamza,and Truc. These papers give some relations between metric completeness and essential self-adjointness. However, [24] contains an example of a graph which is metrically complete in oneof the distances studied in [4] but for which the corresponding weighted Laplacian does not havea unique Markovian extension and is, therefore, not essentially self-adjoint. One reason for thisseems to be that the particular metric used in [4] does not take into account the measure on thevertices of the graph. In [41,42], Milatovic, following [44], shows, with a different metric, thatcompleteness implies essential self-adjointness under the additional assumption of a uniformbound on the vertex degree.

In this paper we investigate these questions for the weighted Laplacian on graphs. Recall thatthe weighted Laplacian has Markovian extensions and the associated form is one of the mostimportant examples of a non-local Dirichlet form. We use the notion of intrinsic distance in-troduced in [11] and show that, if a weighted degree function is bounded on the combinatorialneighborhood of each ball defined with respect to one such distance, then the Laplacian is essen-tially self-adjoint (Theorem 1). As a direct consequence, in the locally finite case, if the graphis metrically complete in one intrinsic path metric, then the Laplacian is essentially self-adjoint(Theorem 2). Compared to the previous results mentioned above we do not assume a uniform

1558 X. Huang et al. / Journal of Functional Analysis 265 (2013) 1556–1578

bound on the vertex degree and, for Theorem 1, we do not even need local finiteness. These re-sults indicate that intrinsic metrics give the correct notion of distance on graphs when seeking toprove statements analogous to the strongly local case.

In the metrically incomplete case, under some further assumptions, we show that if the Cauchy(or metric) boundary has finite capacity, then the Laplacian has a unique Markovian extension ifand only if the Cauchy boundary is polar, that is, has zero capacity, in analogy with [37] (The-orem 3). Moreover, we show that upper Minkowski codimension of the boundary greater than2 implies zero capacity of the boundary (Theorem 4). We also show by examples that the otherimplications do not hold. In particular, in the case when the boundary has infinite capacity, theLaplacian may be essentially self-adjoint or might fail to have a unique Markovian extension, seeExamples 5.2 and 5.4. In general, if the Laplacian is essentially self-adjoint, then it has a uniqueMarkovian extension, but the opposite implication is not necessarily true, see Example 5.1. InExamples 5.5, 5.6, and 5.7 we discuss the case of upper Minkowski codimension less than orequal to 2 where the boundary may be polar or non-polar.

The paper is organized as follows. In Section 2, we introduce the set up, including backgroundmaterial on Dirichlet forms, Laplacians, intrinsic distances, and Cauchy boundary; and statethe main results. In Section 3, we establish the triviality of square integrable eigenfunctionswith negative eigenvalue when the weighted degree function is bounded on the combinatorialneighborhood of each ball and use this to prove Theorems 1 and 2. Section 4 is devoted to theproofs of Theorems 3 and 4 and Section 5 is devoted to (counter-)examples. In Appendix A, weprove a Hopf–Rinow type property for path metrics on locally finite graphs. This property is usedin the proof of Theorems 2 and 3. We also present a series of (counter-)examples showing thatthe property may fail if the graph is not locally finite.

2. The set up and main results

2.1. Weighted graphs

We generally follow the setting of [31]. Let X be a countably infinite discrete set. Elementsof X will be called vertices. A function μ :X → (0,∞) can be viewed as a Radon measure onX with full support so that (X,μ) becomes a measure space.

Let w :X × X → [0,∞) be symmetric, with zero diagonal, and satisfying∑y∈X

w(x, y) < ∞ for all x ∈ X.

The triple (X,w,μ) is called a weighted graph. We call x, y ∈ X neighbors if w(x,y) > 0 anddenote this symmetric relation by x ∼ y. If each vertex has only finitely many neighbors, thenthe graph is called locally finite. For n 1, we call a sequence of points (x0, x1, . . . , xn) a pathconnecting x and y if x0 = x, xn = y, and xi ∼ xi+1 for i = 0,1, . . . , n − 1. A weighted graph(X,w,μ) is called connected if, for any two distinct points in X, there exists a connecting path.From now on, we only consider connected weighted graphs.

2.2. Weighted degree and intrinsic metrics

We call the function Deg :X → [0,∞) given by

Deg(x) := 1

μ(x)

∑y∈X

w(x, y)


the weighted degree. It is, in general, distinct from the combinatorial degree of locally finitegraphs which is given by the number of neighbors of a vertex.

A pseudo metric is a map d :X × X → [0,∞) that is symmetric, has zero diagonal and sat-isfies the triangle inequality. A pseudo metric d = dσ is called a path pseudo metric if there is asymmetric map σ :X × X → [0,∞) such that σ(x, y) > 0 if and only if x ∼ y and

dσ (x, y) = inflσ

((x0, . . . , xn)

) ∣∣ n 1, (x0, . . . , xn) is a path connecting x and y

where the length lσ of a path (x0, . . . , xn) is given by

lσ((x0, . . . , xn)

) =n−1∑i=0

σ(xi, xi+1).

We say that a pseudo metric d has jump size s > 0 if, for all x, y ∈ X, w(x,y) = 0 wheneverd(x, y) > s.

Following Frank, Lenz and Wingert [11] (see Lemma 4.7 and Theorem 7.3) we make a defini-tion which has already proven to be useful in several other problems on graphs, see Remark 2.2below.

Definition. We call a pseudo metric d on (X,w,μ) intrinsic if, for all x ∈ X,

1

μ(x)

∑y∈X

w(x, y)d(x, y)2 1.

An intrinsic path pseudo metric dσ is called strongly intrinsic if, for all x ∈ X,

1

μ(x)

∑y∈X

w(x, y)σ (x, y)2 1.

The first example below shows that there always exist strongly intrinsic path pseudo metricswith jump size 1 on a connected weighted graph.

Example 2.1. (1) For x, y ∈ X with x ∼ y, let σ0(x, y) = minDeg− 12 (x),Deg− 1

2 (y),1. Clearly,dσ0 is strongly intrinsic with jump size 1.

(2) For locally finite graphs, let σ1(x, y) = w(x,y)− 12 min μ(x)

deg(x),

μ(y)deg(y)

12 , x, y ∈ X with

x ∼ y where deg is the combinatorial degree, i.e., the number of neighbors. Clearly, dσ1 is astrongly intrinsic path metric. Moreover, if deg K for some K 1, then dσ1 is equivalent tothe metrics used in [4,5,41,42] (in the case of no magnetic field and no potential). This seems toexplain why the combinatorial vertex degree has to be bounded for these results.

(3) Suppose that σN ≡ 1 on neighbors. Then, dN = dσNgives the natural graph metric, that is,

the distance between x and y is equal to one less than the number of points in the shortest pathconnecting them. Obviously, dN is strongly intrinsic if and only if Deg 1. (Clearly, if Deg isbounded by K > 0, then dN/

√K is also a strongly intrinsic metric.)


Remark 2.2. Various authors came up with such types of metrics independently of [11]. Inthe context of stochastic completeness for jump processes, see the work of Masamune and Ue-mura [40], Grigor’yan, Huang and Masamune [21] and also [26,27]. Independently, Folz [8]came up with similar ideas in the context of heat kernel estimates on locally finite graphs, seealso [9,10]. For further uses of intrinsic metrics, see [25].

For x0 ∈ X and r 0, we define the distance ball with respect to any pseudo metric d byBr(x0) := x ∈ X | d(x, x0) r.

2.3. Forms and operators

In this article, we only consider real valued functions. Denote by C(X) the set of all functionsX → R and by Cc(X) the subset of functions which are finitely supported. The Hilbert spaceL2(X,μ) is defined in the usual way with scalar product

〈u,v〉 :=∑X

uvμ :=∑x∈X

u(x)v(x)μ(x)

and norm ‖u‖ := 〈u,u〉 12 = (

∑X u2μ)

12 .

We next introduce a discrete version of the energy measure which can be thought of as ageneralized gradient. For f ∈ C(X) and x ∈ X define the square of the generalized gradient by

|∇f |2(x) :=∑y∈X

w(x, y)(f (x) − f (y)

)2,

which might take the value ∞. For x ∈ X, let Dloc(x) := (f, g) ∈ C(X) × C(X) |∑y∈X w(x, y)|f (x) − f (y)||g(x) − g(y)| < ∞ and for (f, g) ∈ Dloc(x), we define

(∇f · ∇g)(x) :=∑y∈X

w(x, y)(f (x) − f (y)

)(g(x) − g(y)

).

The generalized form Q is a map C(X) → [0,∞] given by

Q(f ) := 1

2

∑X

|∇f |2 = 1

2

∑x,y∈X

w(x, y)(f (x) − f (y)

)2

and the generalized form domain is given by D := f ∈ C(X) | Q(f ) < ∞. Clearly,Cc(X) ⊆ D. By polarization, this gives a sesqui-linear form Q : D × D → R as follows

Q(f, g) = 1

2

∑X

(∇f · ∇g) = 1

2

∑x,y∈X

w(x, y)(f (x) − f (y)

)(g(x) − g(y)

).

In this context, there are two distinguished restrictions of the generalized form. Let Q be therestriction of Q to

D(Q) := Cc(X)‖·‖Q where ‖·‖Q := (

Q(·) + ‖·‖2) 12 .


The form (Q,D(Q)) is then a regular Dirichlet form, see [31]. Furthermore, let Qmax be therestriction of Q to

D(Qmax) :=

f ∈ L2(X,μ)∣∣ Q(f ) < ∞

.

The form (Qmax,D(Qmax)) is a Dirichlet form but it is not regular in general. For more discus-sion of these two forms and the associated self-adjoint operators in our context, see [24].

The formal Laplacian can be introduced on (X,w,μ) as an analogue of the classicalLaplace–Beltrami operator on Riemannian manifolds as follows

(f )(x) = 1

μ(x)

∑y∈X

w(x, y)(f (x) − f (y)

),

with domain F = f ∈ C(X) | ∑y∈X w(x, y)|f (y)| < ∞ for all x ∈ X. Taking into account∑

y w(x, y) < ∞, x ∈ X, the operator is defined pointwise. It is easy to see that F is stableunder multiplication by bounded functions on X. It can be shown that the self-adjoint operatorL with domain D(L) corresponding to Q is non-negative and is a restriction of , see [31,Theorem 9]. That is,

Lu = u, u ∈ D(L).

2.4. Main results

As discussed in the introduction, it is a classical result that the Laplacian on a weighted man-ifold is essentially self-adjoint if all geodesic balls are relatively compact which is equivalentto the manifold being metrically complete (see, for example, Theorem 11.5 in [20]). Here wepresent some counterparts for weighted graphs.

We define the combinatorial neighborhood n(K) of a subset K of X by

n(K) = x ∈ X | x ∈ K or there exists y ∈ K such that x ∼ y.Note that the combinatorial neighborhood can be understood as the distance one ball about K

with respect to the natural graph distance.

Theorem 1. Let (X,w,μ) be a weighted graph and let d be an intrinsic pseudo metric. If theweighted degree function Deg is bounded on the combinatorial neighborhood of each distanceball, then

D(Q) = D(Qmax), D(L) =

u ∈ L2(X,μ) ∩ F∣∣ u ∈ L2(X,μ)

.

In particular, if additionally Cc(X) ⊆ L2(X,μ), then Lc = |Cc(X) is essentially self-adjoint.

Remark 2.3. (a) The result on essential self-adjointness is sharp by Example 5.1 in Section 5.(b) Let us note that the theorem does not assume that Deg is bounded on X. This would imply

that Q is bounded and the statements become trivial.(c) The condition Cc(X) ⊆ L2(X,μ) holds if and only if w(x, ·)/μ(·) ∈ L2(X,μ) for all

x ∈ X, see Proposition 3.3 in [31]. In particular, this always holds in the locally finite case.


If a pseudo metric has finite jump size, the combinatorial neighborhood of a distance ball iscontained in another distance ball. This yields the following immediate consequence.

Corollary 1. If (X,w,μ) is a weighted graph and d an intrinsic pseudo metric with finite jumpsize such that each distance ball is finite, then the conclusions of Theorem 1 hold.

Note that finite balls and finite jump size imply that the graph is locally finite. In this case,path pseudo metrics are metrics and the analogy to Riemannian manifolds becomes even moreobvious as can be seen below.

Recall that an extension of Lc = |Cc(X) is said to be Markovian, if the form associated to itis a Dirichlet form. In particular, the operators associated to Q and Qmax are Markovian and, inthe locally finite case, having a unique Markovian extension is equivalent to D(Q) = D(Qmax)

[24, Theorem 5.2].

Theorem 2. If (X,w,μ) is a locally finite weighted graph and d = dσ an intrinsic path metricsuch that (X,d) is metrically complete, then

D(L) = u ∈ L2(X,μ)

∣∣ u ∈ L2(X,μ),

Lc = |Cc(X) is essentially self-adjoint and has a unique Markovian extension.

Next, we turn to the metrically incomplete case where we will prove an analogue to resultsfound in [37], see Theorem 3 below. In order to avoid some topological issues when definingcapacity, we now assume that all graphs are locally finite and that we only deal with path metrics.Note that by (a) of Lemma A.3, the topology induced by a path metric is discrete in the locallyfinite case.

For a set U ⊆ X define the capacity of U by

Cap(U) := inf‖u‖Q

∣∣ u ∈ D(Qmax), 1U u

,

where 1U is the characteristic function of U and inf∅ = ∞. For a path metric d on X we let(X,d) be the metric completion. We define the Cauchy boundary ∂CX of X to be the differencebetween X and X:

∂CX := X \ X.

Clearly, X is metrically complete if and only if ∂CX is empty. For A ⊆ X define

Cap(A) := infCap(O ∩ X)

∣∣ A ⊆ O with O ⊆ X open.

Note that, for U ⊆ X, the definitions of capacity agree due to the local finiteness and the use ofpath metrics. We say that ∂CX is polar if Cap(∂CX) = 0.

Theorem 3. Let (X,w,μ) be a locally finite weighted graph and let d = dσ be a strongly intrinsicpath metric. If (X,d) is not metrically complete and Cap(∂CX) < ∞, then the Cauchy boundaryis polar if and only if


D(Q) = D(Qmax),

that is, if and only if Lc = |Cc(X) has a unique Markovian extension.

Note that the other consequences of Theorem 1 do not necessarily follow if (X,d) is metri-cally incomplete with polar boundary. Example 5.1 in Section 5 contains a weighted graph withpolar boundary but where Lc has non-Markovian self-adjoint extensions. Also, in the case whenthe Cauchy boundary has infinite capacity, the Laplacian may be essentially self-adjoint or maynot have a unique Markovian extension, see Examples 5.2 and 5.4.

Next we turn to a criterion which connects polarity of the boundary to codimension of theboundary. The upper Minkowski codimension codimM(∂CX) of ∂CX is defined as

codimM(∂CX) = lim supr→0

lnμ(Br(∂CX))

ln r,

where Br(∂CX) = x ∈ X | infb∈∂CX d(x, b) r. The upper Minkowski codimension (or boxcounting codimension) is one of the most studied fractal dimensions. The relationships betweenvarious dimensions in the classical setting can be found in [7].

Theorem 4. Let (X,w,μ) be a locally finite weighted graph and let d = dσ be an intrinsic pathmetric. If codimM(∂CX) > 2, then ∂CX is polar.

In Examples 5.5, 5.6, and 5.7 we show that ∂CX can be polar or non-polar if codimM(∂CX) 2. See the recent paper [22] for some related statements in the case of manifolds.

3. Uniqueness of solutions

Let (X,w,μ) be a weighted graph and let d be an intrinsic pseudo metric. In this section wewill show that under the assumption that Deg is bounded on the combinatorial neighborhood ofeach distance ball, there are no non-trivial L2 solutions to (+λ)u = 0 for λ > 0. From this factwe can infer Theorems 1 and 2.

3.1. Leibniz rule and Green’s formula

The following auxiliary lemmas are well known in various other situations, see [11,23,24,28,29,31,32]. However, they do not hold on graphs without further assumptions. Here, we provethem under the assumption that the weighted vertex degree is bounded on certain subsets.

Lemma 3.1. Let B ⊆ X be such that Deg is bounded on B . Then, for all u,v ∈ L2(X,μ),

∑x,y∈B

w(x, y)∣∣u(x)v(x)

∣∣ < ∞ and∑

x,y∈B

w(x, y)∣∣u(x)v(y)

∣∣ < ∞.

Proof. Let f = max|u|, |v|. Clearly, f ∈ L2(X,μ). We estimate


∑x,y∈B

w(x, y)∣∣u(x)

∣∣∣∣v(x)∣∣

∑x∈B

f 2(x)∑y∈B

w(x, y) ∑x∈B

f 2(x)μ(x)Deg(x)

supy∈B

Deg(y)‖f ‖2 < ∞,

since Deg is bounded on B by assumption. On the other hand, by the Cauchy–Schwarz inequalityand the above, we obtain

∑x,y∈B

w(x, y)∣∣u(x)

∣∣∣∣v(y)∣∣

( ∑x,y∈B

w(x, y)u2(x)

) 12( ∑

x,y∈B

w(x, y)v2(x)

) 12

< ∞.

The following is an integrated Leibniz rule (see also [11, Theorem 3.7]).

Lemma 3.2 (Leibniz rule). Let U ⊆ X be such that Deg is bounded on n(U). For all f ∈ L∞(X)

with suppf ⊆ U and g,h ∈ L2(X,μ)∑X

(∇(fg) · ∇h) =

∑X

f (∇g · ∇h) +∑X

g(∇f · ∇h).

Proof. By means of Lemma 3.1 and the fact that suppf ⊆ U , it is not hard to see that(fg,h), (f,h) ∈ Dloc(x) for x ∈ n(U), and that (g,h) ∈ Dloc(x) for x ∈ U . Moreover, all ofthe sums over X above are, in fact, sums over n(U) and the first sum on the right-hand sideis over U . Hence, by basic estimates, Lemma 3.1 and the fact suppf ⊆ U , all of the sumsabove converge absolutely. Therefore, the statement follows by the simple algebraic manipula-tion fg(x) − fg(y) = f (x)(g(x) − g(y)) + g(y)(f (x) − f (y)), x, y ∈ X.

The following Green’s formula is a variant of [23, Lemma 4.7].

Lemma 3.3 (Green’s formula). Assume that Deg is bounded on n(U) for some set U ⊆ X. Then,for all u,v ∈ L2(X,μ) ∩ F with suppv ⊆ U ,

∑X

(u)vμ =∑X

u(v)μ = 1

2

∑X

(∇u · ∇v).

Proof. Since w(x,y) = 0 whenever x ∈ U,y ∈ X\n(U), the statement follows by simple alge-braic manipulations, Lemma 3.1 and Fubini’s theorem. 3.2. A Caccioppoli-type inequality

The key estimate for the proof of triviality of L2 solutions to ( + λ)u = 0 for λ > 0 is thefollowing Caccioppoli-type inequality. See [11, Theorem 11.1] for a similar result for generalDirichlet forms.

Lemma 3.4 (Caccioppoli-type inequality). Let u ∈ L2(X,μ) ∩ F , U ⊆ X and assume that Degis bounded on n(U). Then, for all v ∈ L∞(X) with suppv ⊆ U ,

−∑X

(u)uv2μ 1

2

∑X

u2|∇v|2.


Proof. By Lemmas 3.2 and 3.3 we have

∑X

(u)uv2μ = 1

2

∑X

(∇u · ∇(uv2)) = 1

2

∑X

v2|∇u|2 + 1

2

∑X

u(∇u · ∇v2).

We focus on the second sum on the right-hand side. Since a geometric mean can be estimated byits corresponding arithmetic mean, we have |ab| δ

2a2 + 12δ

b2 for a, b ∈ R and δ > 0. We usethis estimate with a = (u(x) − u(y))(v(x) + v(y)), b = u(x)(v(x) − v(y)) and δ = 1/2 for theterms in the second sum on the right-hand side above

∣∣u(x)(u(x) − u(y)

)(v2(x) − v2(y)

)∣∣ 1

4

(u(x) − u(y)

)2(v(x) + v(y)

)2 + u2(x)(v(x) − v(y)

)2

1

2

(u(x) − u(y)

)2(v2(x) + v2(y)

) + u2(x)(v(x) − v(y)

)2.

Multiplying by w(x,y) and summing over x, y ∈ X yields

−1

2

∑X

u(∇u · ∇v2) 1

2

∑X

v2|∇u|2 + 1

2

∑X

u2|∇v|2.

The assertion now follows from the equality in the beginning of the proof. 3.3. Uniqueness of solutions

The following is an analogue of [20, Lemma 11.6].

Proposition 3.5. Assume that the weighted degree function Deg is bounded on the combinatorialneighborhood of each distance ball. Then, for all λ > 0, the equation

( + λ)u = 0,

has only the trivial solution in L2(X,μ) ∩ F .

Proof. Let 0 r < R, fix x0 ∈ X, and consider the cut-off function η = ηR,r :X → R given by

η(x) =(

R − d(x, x0)

R − r

)+

∧ 1.

Note that 0 η 1, η|Br = 1 and η|X\BR= 0. Moreover, η is Lipshitz continuous with Lipshitz

constant 1R−r

(of course, with respect to the intrinsic pseudo metric d). This immediately implies,as d is an intrinsic pseudo metric, that

|∇η|2(x) 1

(R − r)2

∑y∈X

w(x, y)d2(x, y) μ(x)

(R − r)2, x ∈ X.


Fix λ > 0 and assume that u ∈ L2(X,μ) ∩ F is a solution to the equation ( + λ)u = 0. Then,we have by Lemma 3.4 and the estimate on |∇η|2 above, that

λ‖u1Br ‖2 λ‖uη‖2 = −∑X

(u)uη2μ 1

2

∑X

u2|∇η|2 1

2(R − r)2‖u‖2.

Letting R → ∞, we see that u ≡ 0 on Br . Since r is chosen arbitrarily, u ≡ 0 on X. Remark 3.6. It would be interesting to prove a similar statement as Proposition 3.5 for Lp .

3.4. Proofs of Theorems 1 and 2

The proof of Theorem 1 follows by standard techniques used in [31] and [24].

Proof of Theorem 1. By [24, Corollary 4.3], D(Q) = D(Qmax) is equivalent to the non-existence of non-trivial solutions to ( + λ)u = 0 for λ > 0 in D(Qmax). Thus, D(Q) =D(Qmax) follows from Proposition 3.5.

Define Dmax = u ∈ L2(X,μ)∩F | u ∈ L2(X,μ). By Theorem 9 in [31], L is a restrictionof which implies that D(L) ⊆ Dmax. Letting f ∈ Dmax we see that g := (+λ)f ∈ L2(X,μ)

for all λ > 0, so that u := (L + λ)−1g ∈ D(L). As u solves the equation ( + λ)u = g (seeLemma 2.8 in [31]), we conclude that f = u by the uniqueness of solutions, Proposition 3.5 (asf solves the equation by definition). Thus, f ∈ D(L) and, therefore, D(L) = Dmax.

Assuming Cc(X) ⊆ L2(X,μ), essential self-adjointness is a rather immediate consequenceof L = L∗

c . By Green’s formula for functions in v ∈ Cc(X) and f ∈ F (see [23, Lemma 4.7]or [31, Proposition 3.3]), we have

∑X f (Lcv)μ = ∑

X(f )vμ and thus D(L∗c ) = Dmax.

Hence, by what we have shown above, we have D(L∗c ) = D(L) and, therefore, it follows that

L = L∗c .

Proof of Theorem 2. As we assume local finiteness, it is clear that Cc(X) ⊆ L2(X,μ). Fur-thermore, by Theorem A.1, the metric completeness of (X,d) implies that distance balls arefinite. Note that the combinatorial neighborhood of a finite set is again finite. Hence, Degis bounded on the combinatorial neighborhood of each distance ball which implies the state-ments about essential self-adjointness and D(Q) = D(Qmax) by Theorem 1. As uniqueness ofMarkovian extensions is equivalent to D(Q) = D(Qmax) in the locally finite case, see [24, The-orem 4.2], the second statement follows as well. 4. Cauchy boundary and equilibrium potentials

Let (X,w,μ) be a locally finite weighted graph and let d be a path metric. Recall that (X,d)

denotes the metric completion of (X,d) and ∂CX = X \ X denotes the Cauchy boundary. In thissection we prove Theorems 3 and 4.

4.1. Existence of equilibrium potentials

The following is well known and follows directly from [14, Lemma 2.1.1].

Lemma 4.1. If Cap(O) < ∞ for an open set O ⊂ X, then there is a unique element e ∈ D(Qmax)

such that 0 e 1, e|O∩X ≡ 1, and Cap(O) = ‖e‖Q.


Proof. From [14, Lemma 2.1.1] it follows that for any U ⊆ X there is such an e ∈ D(Qmax) (aswe consider X equipped with the discrete topology). Note that in [14, Lemma 2.1.1] regularityof the form is a standing assumption but this is not needed for the proof. Now, for an open setO ⊆ X the equality Cap(O) = Cap(O ∩ X) follows from the definition (by taking A = O).Hence, we let e for O be the corresponding e for O ∩ X.

We call such an e the equilibrium potential associated to O .

4.2. The boundary alternative

The following lemma shows that if the minimal and maximal forms agree, then the capacityof any subset of the boundary is either zero or infinite.

Lemma 4.2. Let A ⊆ ∂CX. If D(Q) = D(Qmax), then either Cap(A) = ∞ or Cap(A) = 0.

Proof. Assume that D(Q) = D(Qmax) and A has finite capacity. Then, there exists an open setO ⊆ X such that A ⊆ O and Cap(O) < ∞. Let e be the equilibrium potential associated to O .Since D(Q) = D(Qmax) there exists a sequence of functions en in Cc(X) converging to e asn → ∞ in the ‖·‖Q norm. Clearly, (e − en)+ ∧ 1 belongs to D(Qmax) and equals 1 on On ∩ X,

where On is a neighborhood of A in X. Therefore,

Cap(A) lim infn→∞ Cap(On) lim

n→∞∥∥(e − en)+ ∧ 1

∥∥Q

limn→∞‖e − en‖Q = 0.

4.3. Approximation by equilibrium potentials

Next, we show that every bounded function in D(Qmax) can be approximated via equilibriumpotentials if the boundary has capacity zero.

Lemma 4.3. Assume that ∂CX is polar and let en be the equilibrium potentials associated toopen sets On ⊆ X with ∂CX ⊆ On and Cap(On) → 0 as n → ∞. Then ‖u − (1 − en)u‖Q → 0as n → ∞ for all u ∈ D(Qmax) ∩ L∞(X).

Proof. Note that u − (1 − en)u = enu and ‖enu‖ ‖u‖∞‖en‖ → 0 as n → ∞. Moreover, wehave

Q(enu) = 1

2

∑X

∣∣∇(enu)∣∣2

∑X

e2n|∇u|2 +

∑X

u2|∇en|2

∑X

e2n|∇u|2 + 2‖u‖2∞Q(en) → 0 as n → ∞

by noting that en(x) → 0 for all x and applying the Lebesgue dominated convergence theo-rem.


4.4. Restriction to complete subgraphs

In the next lemma we show that bounded functions in D(Qmax) that are zero close to theboundary can be approximated by finitely supported functions. We show this by restricting ourattention to complete subgraphs. In order for the restriction of an intrinsic metric to be intrinsic,we need to assume that the metric is strongly intrinsic.

Lemma 4.4. Assume that d = dσ is a strongly intrinsic path metric. Let O ⊆ X be open with∂CX ⊆ O , Cap(O) < ∞ and let e be the equilibrium potential associated to O . Then, Cc(X) isdense in (1−e)(D(Qmax)∩L∞(X)) = (1−e)u | u ∈ D(Qmax)∩L∞(X) with respect to ‖·‖Q.

Proof. Let Y = X \O , μY be the restriction of μ to Y and σY and wY be the restrictions of σ andw to Y × Y . We first assume that (Y,wY ,μY ) is connected. From this it follows that dY = dσY

isa strongly intrinsic path metric on (Y,wY ,μY ) with dY d .

Claim. (Y, dY ) is metrically complete.

Proof. If (xn) is a Cauchy sequence in Y , then, by dY d , it is a Cauchy sequence in X and hasa limit point in X. However, as Y = X \ O , the limit point is not in ∂CX. As (X,dσ ) is a discretemetric space, see Lemma A.3(a), (xn) must be eventually constant which proves the claim.

For R > 0 and fixed x0 ∈ Y , let ηR :Y → [0,1] given by

ηR(x) =(

2R − dY (x, x0)

R

)+

∧ 1.

By completeness, B2R in (Y, dY ) is finite, see Theorem A.1 in Appendix A, and it follows thatηR(1 − e)D(Qmax) ⊆ Cc(X).

Let ∇ be the generalized gradient for (Y,wY ,μY ). Let v ∈ (1 − e)(D(Qmax) ∩ L∞(X)) andset gR = v − ηRv = (1 − ηR)v. Now,

Q(gR) = 1

2

∑Y

|∇gR|2 +∑x∈Y

∑y∈O∩X

w(x, y)g2R(x).

For the first term we get, using that dY is intrinsic and, therefore, that |∇ηR|2 μY /R2,

1

2

∑Y

|∇gR|2 ∑Y

v2|∇ηR|2 +∑Y

(1 − ηR)2|∇v|2 1

R2‖v‖2 +

∑Y

(1 − ηR)2|∇v|2 → 0,

as R → ∞. For the second term let u ∈ (D(Qmax) ∩ L∞(X)) such that v = (1 − e)u. Then,

∑x∈Y

∑y∈O∩X

w(x, y)gR(x)2 ‖u‖2∞∑x∈Y

(1 − ηR(x)

)2 ∑y∈O∩X

w(x, y)(1 − e(x)

)2 → 0

as R → ∞ by the Lebesgue dominated convergence theorem. This follows, since ηR → 1 point-wise as R → ∞ and

∑Y

∑O∩X w(x, y)(1 − e(x))2 Q(e) Cap(O)2 < ∞ (as e(y) = 1 for

y ∈ O ∩ X). Moreover, as gR converges pointwise to zero it also converges to zero in L2.


In the case where Y is not connected there are at most countably many connected compo-nents Yi , i 0. For v ∈ (1 − e)(D(Qmax) ∩ L∞(X)) let vi = v|Yi

. Since Q(v) = ∑i0 Q(vi),

the statement follows by a diagonal sequence argument. 4.5. Proof of Theorem 3

Proof of Theorem 3. Since we assume local finiteness, Lc having a unique Markovian extensionis equivalent to D(Q) = D(Qmax) by Theorem 5.2 in [24].

If D(Q) = D(Qmax), then the assumption Cap(∂CX) < ∞ implies Cap(∂CX) = 0 byLemma 4.2.

If, on the other hand, ∂CX has zero capacity, then Cc(X) is dense in D(Qmax) ∩ L∞(X)

with respect to ‖·‖Q by Lemmas 4.3 and 4.4. But D(Qmax) ∩ L∞(X) is dense in D(Qmax) withrespect to ‖·‖Q since if u ∈ D(Qmax), then un = (u∨−n)∧ n converges to u in ‖·‖Q as n → ∞(cf. [14, Theorem 1.4.2(iii)]). This implies D(Q) = D(Qmax). 4.6. Proof of Theorem 4

Proof of Theorem 4. As codimM(∂CX) > 2, there exists an ε > 0 and a sequence rn → 0 asn → ∞ such that

μ(Brn(∂CX)

)< r2+ε

n .

For R > 0 and x ∈ X, let

ηR(x) =(

2R − d(x, ∂CX)

R

)+

∧ 1.

In particular, 0 ηR 1, ηR|BR(∂CX) ≡ 1 and ηR|X\B2R(∂CX) ≡ 0. It follows that

‖ηR‖2 μ(B2R(∂CX)

)and

Q(ηR) ∑

x∈B2R(∂CX)

∑y∈X

w(x, y)

(d(x, ∂CX)

R− d(y, ∂CX)

R

)2

1

R2

∑x∈B2R(∂CX)

∑y∈X

w(x, y)d(x, y)2 μ(B2R(∂CX))

R2

since d is intrinsic.Applying the above with rn/2 in place of R, it follows that

Cap(∂CX) ‖ηrn/2‖Q (

μ(Brn(∂CX)

) + 4

r2n

μ(Brn(∂CX)

)) 12

(r2+εn + 4rε

n

) 12 → 0 as n → ∞.


5. (Counter-)examples

Here we present the examples mentioned in Section 2.4. In particular, we show that Markovuniqueness does not imply essential self-adjointness, that no conclusion can be drawn concerninguniqueness in the infinite capacity case and that the boundary can be polar or non-polar for anyupper Minkowski codimension less than or equal to 2.

As we often use a graph with X = N0 and x ∼ y if and only if |x − y| = 1 we make severalpreliminary observations concerning graphs of this type with a given path metric d . First, in thiscase,

∂CX = ∅ if and only if l(X) :=∞∑

x=0

d(x, x + 1) < ∞,

see Theorem A.1 in Appendix A. Second, if ∂CX = ∅, then

Cap(∂CX) < ∞ if and only if μ(X) < ∞.

This can be seen as follows: if μ(X) = ∞, then every neighborhood of the boundary must haveinfinite measure so that Cap(∂CX) = ∞. If μ(X) < ∞, then 1 ∈ D(Qmax) which implies thatCap(∂CX) ‖1‖Q = μ(X) < ∞. These two observations will be used repeatedly below.

Example 5.1. (Polar Cauchy boundary (and consequently D(Q) = D(Qmax)) but no es-sential selfadjointness.) Let X = Z with w(x,y) = 1 if |x − y| = 1 and 0 otherwise. Thestrongly intrinsic path metric d = dσ0 introduced in Example 2.1 satisfies d(x, x + 1) =min√μ(x)/2,

√μ(x + 1)/2,1 for an arbitrary measure μ. Therefore, if the measure is cho-

sen so that it satisfies∑∞

x=−∞ x2√μ(x) < ∞, then (X,d) is metrically incomplete, the Cauchyboundary consists of two points and h : x → x is in L2(X,μ) which we will use later.

Define en by

en(x) := (|x|/n − 1)+ ∧ 1.

One checks that en ∈ D(Qmax) with

Q(en) =∞∑

x=−∞

(en(x) − en(x + 1)

)2 = 2n1

n2→ 0

and that en → 0 in L2(X,μ) as n → ∞ by the Lebesgue dominated convergence theorem. Thus,the Cauchy boundary of X is polar and Lc = |Cc(X) has a unique Markovian extension.

On the other hand, the formal Laplacian acts as f (x) = 1μ(x)

(f (x) − f (x − 1) + f (x) −f (x+1)). Clearly, h(x) = x is harmonic, square integrable by the choice of μ, and h /∈ D(Qmax).This shows that h ∈ D(L∗

c ) \ D(Qmax), that is, Lc is not essentially self-adjoint.

Example 5.2. (Cauchy boundary with infinite capacity and essential self-adjointness.) Let X =N0 with μ(X) = ∞ and w symmetric such that w(x,y) > 0 if and only if |x − y| = 1. By [31,Theorem 6] the operator |Cc(X) is essentially self-adjoint. (This can be also seen directly asthere are no non-trivial solutions to ( + λ)u = 0 in L2(X,μ) for λ > 0. This follows as any


positive solution to this equation must be increasing by a minimum principle, see also Eq. (1)below.)

If d = dσ0 and w and μ are chosen to satisfy

l(X) = limx→∞d(0, x)

∞∑x=0

(1

Deg(x)

) 12

∞∑

x=0

(μ(x)

w(x, x + 1)

) 12

< ∞,

then it follows that (X,d) is not metrically complete and that the boundary consists of a singlepoint. Since μ(X) = ∞, Cap(∂CX) = ∞ as noted above.

Example 5.3. (Cauchy boundary with finite positive capacity and consequently D(Q) =D(Qmax).) Let X = N0 with μ(X) < ∞ and let w be symmetric with w(x,y) > 0 if and only if|x − y| = 1 and satisfying

l(X) = limx→∞d(0, x)

∞∑x=0

(μ(x)

w(x, x + 1)

) 12

< ∞ and∞∑

x=0

1

w(x,x + 1)< ∞

where d = dσ0 . In particular, the Cauchy boundary ∂CX of X consists of one point and has finitecapacity.

Recall that D(Q) = D(Qmax) is equivalent to ( + λ)u = 0 having a non-trivial solution inD(Qmax) for any λ > 0 [24, Corollary 4.3]. By [33, Lemma 4.3], the equation ( + λ)u = 0 onX translates to

u(x + 1) − u(x) = λ

w(x, x + 1)

x∑y=0

u(y)μ(y) (1)

from which it follows, see [33, Lemma 5.4], that u is bounded if and only if

∞∑x=0

∑xy=0 μ(y)

w(x, x + 1)< ∞.

As μ(X) < ∞, this is equivalent to

∞∑x=0

1

w(x,x + 1)< ∞.

Furthermore, as μ(X) < ∞, u ∈ L∞(X) implies that u ∈ L2(X,μ). It is also not difficult to seethat Q(u) < ∞ in this case as, by (1), we get that

w(x,x + 1)(u(x + 1) − u(x)

)2 1

w(x,x + 1)

(λμ(X)‖u‖∞

)2.

Therefore, as u ∈ D(Qmax) is non-trivial, D(Q) = D(Qmax). Finally, Cap(∂CX) > 0 follows bycombining Cap(∂CX) < ∞, D(Q) = D(Qmax), and Theorem 3.


Example 5.4. (Cauchy boundary with infinite capacity and D(Q) = D(Qmax) (and consequentlyno essential self-adjointness).) We consider X = Z with X = X− ∪ X+ where X− = −N0 withw and μ chosen as in Example 5.2 and X+ = N0 with w and μ chosen as in Example 5.3. Inparticular, the Cauchy boundary ∂CX of X consists of two points, pL and pR , and has infinitecapacity as Cap(pL) = ∞ by μ(X−) = ∞. On the other hand, by Example 5.3 we have 0 <

Cap(pR) < ∞ which gives D(Q) = D(Qmax) by Lemma 4.2.

Example 5.5. (Polar Cauchy boundary with upper Minkowski codimension 2.) Let X = N0with w(x,y) = 1/8 if |x − y| = 1 and 0 otherwise and μ(x) = 4−x . Therefore, for x > 0,Deg(x) = 4x−1 so that, with d = dσ0 , we get d(x, x + 1) = 2−x . Furthermore, by using thetechnique of Example 5.1, we can show that the Cauchy boundary consists of a single point, pR ,and that Cap(pR) = 0. Let r(x) := d(x,pR) = ∑∞

y=x 2−y = 2−(x−1) so that μ(Br(x)(pR)) =∑∞y=x 4−y = 4−(x−1)/3 = r(x)2/3. Therefore,

lnμ(Br(x)(pR))

ln r(x)= 2 ln r(x) − ln 3

ln r(x)→ 2 as x → ∞.

Example 5.6. (Non-polar Cauchy boundary with upper Minkowski codimension 2.) Let X = N0,with w symmetric, satisfying w(x,x + 1) = (x + 1)2 and 0 otherwise with

d(x, x + 1) = 1

2x+2and μ(x) = (x + 1)2

4x.

It is easy to check that this metric is intrinsic. As l(X) < ∞, μ(X) < ∞, and∑∞

x=01

w(x,x+1)<

∞ it follows that

0 < Cap(∂CX) < ∞

by the reasoning of Example 5.3. Therefore, codimM(∂CX) 2 by Theorem 4. By definition,

r(x) := d(x, ∂CX) = 1

2x+1and μ

(Br(x)(∂CX)

) =∞∑

y=x

(y + 1)2

4y.

Now, for every β > 1/4, there exists an M such that (x + 1)2 (4β)x for all x M . Hence, forall x M and 1/4 < β < 1, we have μ(Br(x)(∂CX))

∑∞y=x βy = βx(1 − β)−1. Therefore, for

all x M ,

lnμ(Br(x)(∂CX))

ln r(x) ln(βx(1 − β)−1)

ln 2−(x+1)

which implies that codimM(∂CX) −lnβ/ln 2. As β > 1/4 was chosen arbitrarily, it followsthat codimM(∂CX) 2 yielding that codimM(∂CX) = 2.

Example 5.7. (Upper Minkowski codimension between 0 and 2 with polar and non-polar Cauchyboundary.) Let X = N0 with w(x,y) > 0 if and only if |x − y| = 1. Let, for α ∈ R,


d(x, x + 1) = 1

2α(x+1)and μ(x) = 1

2(2α−1)x.

Then, l(X) < ∞ for α > 0 and μ(X) < ∞ for α > 1/2. Thus, Cap(∂CX) < ∞ for α > 1/2.Now,

r(x) := d(x, ∂CX) =∞∑

y=x

1

2α(y+1)=

(1

2α − 1

)1

2αx

and

μ(Br(x)(∂CX)

) =∞∑

y=x

1

2(2α−1)y=

(1

22α−1 − 1

)1

2(2α−1)(x−1)

so that

codimM(∂CX) = 2 − 1

α.

We now specify two choices of weights w:Case 1 – polar Cauchy boundary: Let w(x,x + 1) = 1 for all x ∈ N0. Clearly, d is intrinsic

for all α > 0. Furthermore, if Cap(∂CX) < ∞, then Cap(∂CX) = 0 as in Example 5.1. Hence,there exist examples of graphs with polar Cauchy boundary such that 0 < codimM(∂CX) < 2.

Case 2 – non-polar Cauchy boundary: Let w(x,x + 1) = 2x for all x ∈ N0. It is easy tosee that d is intrinsic for all α 1

2 . Furthermore, since∑∞

x=01

w(x,x+1)< ∞ one can show that

Cap(∂CX) > 0 as in Example 5.3. Consequently, there exists a family of graphs with non-polarCauchy boundary such that 0 < codimM(∂CX) < 2.

Acknowledgments

M.K. enjoyed various inspiring discussion with Daniel Lenz and gratefully acknowledges thefinancial support from the German Research Foundation (DFG). R.K.W. thanks Józef Dodz-iuk for numerous insights and acknowledges the financial support of the FCT under projectPTDC/MAT/101007/2008 and of the PSC-CUNY Awards, jointly funded by the ProfessionalStaff Congress and the City University of New York. The authors are grateful to Ognjen Mila-tovic for a careful reading of the manuscript and suggestions.

Appendix A. A Hopf–Rinow type theorem

Let (X,w,μ) be a weighted graph.A metric space (X,d) is said to be metrically complete if every Cauchy sequence converges to

an element in X. A path (xn) (finite or infinite) is called a geodesic with respect to a path metricd = dσ if d(x0, xn) = lσ ((x0, . . . , xn)) for all n > 0. A weighted graph (X,w,μ) with a pathmetric d = dσ is said to be geodesically complete if all infinite geodesics have infinite lengths,i.e., lσ ((xk)) = limn→∞ lσ ((x0, . . . , xn)) = ∞ for all infinite geodesics (xk).


We prove the following Hopf–Rinow type theorem.

Theorem A.1. Let (X,w,μ) be a locally finite weighted graph and d be a path pseudo metric.Then, (X,d) is a discrete metric space. Moreover, the following are equivalent:

(i) (X,d) is metrically complete.(ii) (X,d) is geodesically complete.

(iii) Every distance ball is finite.(iv) Every bounded and closed set is compact.

In particular, if (X,d) is complete, then for all x, y ∈ X there is a path (x0, . . . , xn) connectingx and y such that dσ (x, y) = lσ ((x0, . . . , xn)).

Remark A.2. (a) Anytime a path pseudo metric d induces the discrete topology on X the follow-ing implications hold: (iii) ⇔ (iv) ⇒ (i) ⇒ (ii). This is the case if and only if infy∼x σ (x, y) > 0for all x ∈ X. In fact, (iv) ⇒ (i) holds for general metric spaces. The stronger assumption of localfiniteness is needed for the implications (ii) ⇒ (i), (i) ⇒ (iii) (or (iv)) and (ii) ⇒ (iii) (or (iv)).See Example A.5 below.

(b) A similar statement as (i) ⇒ (iii) can also be found in [41].

We prove the theorem in several steps through the following lemmas.

Lemma A.3. Let (X,w,μ) be a locally finite weighted graph and d be a path pseudo metric.Then, the following hold:

(a) (X,d) is a discrete metric space. In particular, (X,d) is locally compact.(b) A set is compact in (X,d) if and only if it is finite.

Proof. Local finiteness and the assumption σ(x, y) > 0 for x ∼ y, imply that for all x ∈ X thereis an r > 0 such that d(x, y) > r for all y ∈ X with y ∼ x. First, by the definition of d , we havethat for all x, z ∈ X with x = y there is y ∼ x with d(x, y) d(x, z). Thus, d(x, y) = 0 impliesx = y, therefore, d is a metric. Second, it yields that Br(x) = x and x is an open set whichshows (a). From this we conclude that for any infinite set U the cover x | x ∈ U has no finitesubcover. The other direction of (b) is clear.

The authors are grateful to Florentin Münch for a crucial idea in the proof of the followinglemma.

Lemma A.4. Let (X,w,μ) be a locally finite weighted graph and d be a path metric. Assumethat Br is infinite for some r 0. Then, there exists an infinite geodesic of bounded length.

Proof. Let o ∈ X be the center of the infinite ball Br of radius r and let dN be the natural graphdistance. Let Pn, n > 0, be the set of finite paths (x0, . . . , xk) such that x0 = o, xi = xj for i = j ,dN(xk, o) = n and dN(xj , o) n for j = 0, . . . , k.

Claim. Γn = γ ∈ Pn | γ geodesic with respect to d, l(γ ) r = ∅ for all n > 0.


Proof. The set Pn is finite by local finiteness of the graph and, thus, contains a minimal elementγ = (x0, . . . , xK) with respect to the length l, i.e., for all γ ′ ∈ Pn we have l(γ ′) l(γ ). Then,γ is a geodesic: for every path (x′

0, . . . , x′M) with x′

0 = o and x′M = xK , we let m ∈ n, . . . ,M

be such that (x′0, . . . , x

′m) ∈ Pn. By the minimality of γ we infer

l((

x′0, . . . , x

′M

)) l

((x′

0, . . . , x′m

)) l(γ ).

It follows that γ is a geodesic. Clearly, l(γ ) r , as otherwise Br ⊆ y ∈ X | dN(y, o) n − 1which would imply the finiteness of Br by the local finiteness of the graph. Thus, γ ∈ Γn whichproves the claim.

We inductively construct an infinite geodesic (xk) with bounded length: We set x0 = o. SinceΓn = ∅, there is a geodesic in Γn for every n > 0 which contains x0. Suppose we have constructeda geodesic (x0, . . . , xm) such that for all n m there is a geodesic in Γn that has (x0, . . . , xm)

as a subgeodesic. By local finiteness xm has finitely many neighbors. Thus, there must be aneighbor xm+1 of xm such that for infinitely many n the path (x0, . . . , xm, xm+1) is a subpathof a geodesic in Γn. However, a subpath of a geodesic is a geodesic. Thus, there is an infi-nite geodesic γ = (xk)k0 with l(γ ) = limn→∞ l((x0, . . . , xn)) r as (x0, . . . , xn) ∈ Γn for alln 0. Proof of Theorem A.1. The fact that (X,d) is a discrete metric space follows from Lemma A.3.We now turn to the proof of the equivalences. We start with (i) ⇒ (ii). If there is a boundedinfinite geodesic, then it is a Cauchy sequence. Since a geodesic is a path it is not eventuallyconstant, thus it does not converge by discreteness. Hence, (X,d) is not metrically complete.To prove (ii) ⇒ (iii) suppose that there is a distance ball that is infinite. By Lemma A.4 thereis a bounded infinite geodesic and (X,d) is not geodesically complete. From Lemma A.3(b) wededuce (iii) ⇔ (iv). Finally, we consider the direction (iv) ⇒ (i). If every bounded and closed setis compact, then every closed distance ball is compact. Then, by Lemma A.3(b) every distanceball is finite and it follows that (X,d) is metrically complete.

We finish this appendix by giving several (counter-)examples to show that some of the state-ments above fail to be true in the case of non-locally finite graphs. We present the examples withrespect to the path metric with σ = σ0 (see Example 2.1). Another example of this type can befound in [11, Example 14.1].

Example A.5. Let μ ≡ 1, σ = σ0, and d = dσ .(1) A metrically and geodesically complete graph with non-compact distance balls.This example can be thought of as a star graph, where the rays are two subsequent edges.

Let X = N0 and let w be symmetric with w(0,2n) = 1/2n and w(2n − 1,2n) = 1 − 1/2n forn ∈ N and w ≡ 0 otherwise. We have d(0,2n) = 1 for n ∈ N. Then, (X,d) is metrically (andgeodesically) complete but B1(0) is not compact.

(2) A non-locally compact graph.This example can be thought of as a star graph where the rays are copies of N whose

lengths become shorter. Let X = N20 and let w be symmetric with w((0,0), (m,0)) = 1/2m

for m ∈ N and w((m,n − 1), (m,n)) = 22(m+n)/5 for m,n ∈ N and w ≡ 0 otherwise. Then,Deg((0,0)) = 1, Deg((m,0)) = 1/2m + 22(m+1)/5 and Deg((m,n)) = 22(m+n) for m,n ∈ N.Hence, we have d((m,n− 1), (m,n)) = 2−(m+n) for m,n ∈ N and 1/2m+1 d((0,0), (m,n))


3/2m+1. For a ball Br((m,n)), m,n 0, r > 0, denote by Ur((m,n)) its interior. Now for ε > 0,Uε/2((0,0))∪ U1/2(m+n+1) ((m,n)) | m 1, n 0 is an open cover of Bε((0,0)) with no finitesubcover.

(3) A non-Hausdorff space.This example can be thought as two vertices which are connected by infinitely many paths that

become shorter. Let X = N0 ∪ ∞ and let w be symmetric with w(0,2n) = w(∞,2n) = 1/2n

and w(2n − 1,2n) = 22n and w(n,m) = 0 all other m,n ∈ N0. Then, σ(0,2n) = σ(∞,2n) 1/2n. Hence, d(0,∞) = 0.

(4) An infinite ball and non-discreteness.This example is a modification of (1). Let X = N0 and let w be symmetric with w(0,2n) =

1/2n and w(2n − 1,2n) = 2n and w(n,m) = 0 all other m,n ∈ N0. Then, every d-ball about 0is compact but it contains infinitely many vertices. Moreover, the vertices xn = 2n converge tox = 0 with respect to d . (This is, in particular, a counterexample to Lemma A.3 for non-locallyfinite graphs.)

(5) A geodesically complete graph which is not metrically complete.This example is an extension of (3) and can be thought as a “line graph” where between

each two points on the line there are infinitely many “line segments” that become shorter. LetX = N2

0 and let w be symmetric with w((m,0), (m,2n)) = 1/2n = w((m + 1,0), (m,2n)) andw((m,2n), (m,2n − 1)) = 22(m+1) − 3/2n for m ∈ N0, n ∈ N and w ≡ 0 otherwise. It fol-lows that Deg((m,2n)) = 22(m+1) − 1/2n implying that d((m,0), (m + 1,0)) = 1/2m. Thus(xm) = ((m,0)) is a Cauchy sequence which does not converge. On the other hand, the spaceis geodesically complete as there are no infinite geodesics.

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CHAPTER 10

F. Bauer, M. Keller, R. Wojciechowski, Cheegerinequalities for unbounded graph Laplacians, to

appear in Journal of the European MathematicalSociety.

239

CHEEGER INEQUALITIES FOR UNBOUNDEDGRAPH LAPLACIANS

FRANK BAUER, MATTHIAS KELLER,AND RADOS LAW K. WOJCIECHOWSKI

Abstract. We use the concept of intrinsic metrics to give a newdefinition for an isoperimetric constant of a graph. We use thisnovel isoperimetric constant to prove a Cheeger-type estimate forthe bottom of the spectrum which is nontrivial even if the vertexdegrees are unbounded.

1. Introduction

In 1984 Dodziuk [9] proved a lower bound on the spectrum of theLaplacian on infinite graphs in terms of an isoperimetric constant.Dodziuk’s bound is an analogue of Cheeger’s inequality for manifolds[6] except for the fact that Dodziuk’s estimate also contains an upperbound for the vertex degrees in the denominator. In a later paper [12]Dodziuk and Kendall expressed that it would be desirable to have anestimate without the rather unnatural vertex degree bound. They over-came this problem in [12] by considering the normalized Laplace opera-tor, which is always a bounded operator, instead. However, the originalproblem of finding a lower bound on the spectrum of unbounded graphLaplace operators that only depends on an isoperimetric constant re-mained open until today.

In this paper, we solve this problem by using the concept of intrinsicmetrics. More precisely, for a given weighted Laplacian, we use an in-trinsic metric to redefine the boundary measure of a set. This leads toa modified definition of the isoperimetric constant for which we obtaina lower bound on the spectrum that depends solely on the constant.These estimates hold for all weighted Laplacians (including boundedand unbounded Laplace operators). The strategy of proof is not sur-prising, as it does not differ much from the one of [9, 12]. However, themain contribution of this note is to provide the right definition of anisoperimetric constant to solve the open problem mentioned above.

Date: March 17, 2013.1

2 F. BAUER, M. KELLER, AND R. WOJCIECHOWSKI

To this day, there is a vital interest in estimates of isoperimetric con-stants and in Cheeger-type inequalities. For example, rather classicalestimates for isoperimetric constants in terms of the vertex degree canbe found in [2, 11, 43, 44]; for relations to random walks, see [21, 48].While, for regular planar tessellations, isoperimetric constants can becomputed explicitly [26, 28]; for arbitrary planar tessellations there areestimates in terms of curvature [27, 35, 40, 47]. For Cheeger inequal-ities on simplicial complexes, there is recent work found in [45]; forgeneral weighted graphs, see [10, 37]. Moreover, Cheeger estimates forthe bottom of the essential spectrum and criteria for discreteness ofspectrum are given in [19, 34, 49, 50]. Upper bounds for the top of the(essential) spectrum and another criterium for the concentration of theessential spectrum in terms of the dual Cheeger constant are given in[3]. Finally, let us mention works connecting discrete and continuousCheeger estimates [1, 8, 33, 41].

The paper is structured as follows. The set up is introduced in thenext section. The Cheeger inequalities are presented and proven inSection 3. Moreover, upper bounds are discussed. A technique toincorporate non-negative potentials into the estimate is discussed inSection 4. Section 5 is dedicated to relating the exponential volumegrowth of a graph to the isoperimetric constant via upper bounds whilelower bounds on the isoperimetric constant in the flavor of curvatureare presented in Section 6. These lower bounds allow us to give exam-ples where our estimate yields better results than all estimates knownbefore.

2. The set up

2.1. Graphs. Let X be a countably infinite set equipped with thediscrete topology. A function m : X → (0,∞) gives a Radon measureon X of full support via m(A) =

∑x∈Am(x) for A ⊆ X, so that (X,m)

becomes a discrete measure space.A graph over (X,m) is a symmetric function b : X × X → [0,∞)

with zero diagonal that satisfies∑

y∈Xb(x, y) <∞ for x ∈ X.

We can think of x and y as neighbors, i.e, being connected by an edge,if b(x, y) > 0 and we write x ∼ y. In this case, b(x, y) is the strengthof the bond interaction between x and y. For convenience we assumethat there are no isolated vertices, i.e., every vertex has a neighbor.We call b locally finite if each vertex has only finitely many neighbors.

CHEEGER INEQUALITIES FOR UNBOUNDED GRAPH LAPLACIANS 3

The measure n : X → (0,∞) given by

n(x) =∑

y∈Xb(x, y) for x ∈ X.

plays a distinguished role in the proof of classical Cheeger inequalities.In the case when b : X×X → 0, 1, n(x) gives the number of neighborsof a vertex x.

2.2. Intrinsic metrics. We call a pseudo metric d for a graph b on(X,m) an intrinsic metric if

∑

y∈Xb(x, y)d(x, y)2 ≤ m(x) for all x ∈ X.

The concept of intrinsic metrics was first studied systematically bySturm [46] for strongly local regular Dirichlet forms and it was gen-eralized to all regular Dirichlet forms by Frank/Lenz/Wingert in [17].By [17, Lemma 4.7, Theorem 7.3] it can be seen that our definition co-incides with the one of [17]. A possible choice for d is the path pseudo

metric induced by the edge weights w(x, y) = ((m/n)(x)∧ (m/n)(y))12 ,

for x ∼ y, see e.g. [30]. Moreover, the natural graph metric (i.e., thepath metric with weights w(x, y) = 1 for x ∼ y) is intrinsic if m ≥ n.Intrinsic metrics for graphs were recently discovered independently invarious contexts, see e.g. [14, 15, 16, 23, 25, 30, 31, 32, 42], where cer-tain variations of the concept also go under the name adapted metrics.

2.3. Isoperimetric constant. In this section we use the concept ofintrinsic metrics to give a refined definition of the isoperimetric con-stant. As it turns out, this novel isoperimetric constant is more suitablethan the classical one if n ≥ m. Let W ⊆ X. We define the boundary∂W of W by

∂W = (x, y) ∈ W ×X \W | b(x, y) > 0.For a given intrinsic metric d we set the measure of the boundary as

|∂W | =∑

(x,y)∈∂Wb(x, y)d(x, y).

Note that |∂W | <∞ for finite W ⊆ X by the Cauchy-Schwarz inequal-ity and the assumption that

∑y b(x, y) < ∞. We define the isoperi-

metric constant or Cheeger constant α(U) = αd,m(U) for U ⊆ X as

α(U) = infW⊆Ufinite

|∂W |m(W )

.


If U = X, we write

α = α(X).

For b : X × X → 0, 1 and d the natural graph metric the measureof the boundary |∂W | is number of edges leaving W . If additionallym = n, then our definition of α coincides with the classical one from[12].

2.4. Graph Laplacians. Denote by Cc(X) the space of real valuedfunctions on X with compact support. Let `2(X,m) be the spaceof square summable real valued functions on X with respect to themeasure m which comes equipped with the scalar product 〈u, v〉 =∑

x∈X u(x)v(x)m(x) and the norm ‖u‖ = ‖u‖m = 〈u, u〉 12 . Let theform Q = Qb with domain D be given by

Q(u) =1

2

∑

x,y∈Xb(x, y)(u(x)− u(y))2, D = Cc(X)

‖·‖Q,

where ‖ · ‖Q = (Q(·) + ‖ · ‖2)12 . The form Q defines a regular Dirichlet

form on `2(X,m), see [20, 36]. The corresponding positive selfadjointoperator L can be seen to act as

Lf(x) =1

m(x)

∑

y∈Xb(x, y)(f(x)− f(y)),

(cf. [36, Theorem 9]). Let L be the extension of L to F = f : X →R |∑y∈X b(x, y)|f(y)| <∞ for all x ∈ X. We have Cc(X) ⊆ D(L) if

(and only if) LCc(X) ⊆ `2(X,m), see [36, Theorem 6]. In particular,this can easily seen to be the case if the graph is locally finite or ifinfx∈X m(x) > 0. Note that L becomes a bounded operator if and onlyif Cm ≥ n for some C > 0 (cf. [24, Theorem 9.3]). In particular, ifm = n, then L is referred to as the normalized Laplacian.

We denote the bottom of the spectrum σ(L) and the essential spec-trum σess(L) of L by

λ0(L) = inf σ(L) and λess0 (L) = inf σess(L).

3. Cheeger inequalities

Let b be a graph over (X,m) and d be an intrinsic metric. In thissection we prove the main results of the paper.


3.1. Main results.

Theorem 3.1.

λ0(L) ≥ α2

2.

Remark. (a) A similar statement can be proven for weighted Lapla-cians on finite graphs for the first non-zero eigenvalue. In this case, theinfimum in the definition of the isoperimetric constant has to be takenover all sets that have at most half of the measure of the whole graph.The main part of the proof works similarly, for details of the adaptionto the finite graph case, see [7, proof of Theorem 2.2].

(b) We also obtain a similar statement for the self adjoint operatorthat is related to the maximal form (cf. Section 5) which is discussedunder the name Neumann Laplacian in [24]. Here one has to take theinfimum in the definition of the isoperimetric constant over all sets offinite measure. With this choice, all of our proofs work analogously.

Under the additional assumption that L is bounded with operatornorm 1, we recover the classical Cheeger inequality from [19, 43] whichcan be seen to be stronger than the one of [12] by the Taylor expansion.We say that d ≥ 1 (respectively d ≤ 1) for neighbors if d(x, y) ≥ 1(respectively d(x, y) ≤ 1) for all x ∼ y.

Theorem 3.2. If m ≥ n and d ≥ 1 or d ≤ 1 for neighbors, then

λ0(L) ≥ 1−√

1− α2.

In order to estimate the essential spectrum let the isoperimetric con-stant at infinity be given by

α∞ = supK⊆Xfinite

α(X \K),

which coincides with the one of [37] in the case of the natural graphmetric and with the one of [19, 34] if additionally b : X ×X → 0, 1and m = n. Note that the assumptions of the following theorem are inparticular fulfilled if the graph is locally finite or if infx∈X m(x) > 0.

Theorem 3.3. Assume Cc(X) ⊆ D(L). Then,

λess0 (L) ≥ α2

∞2.

3.2. Co-area formulae. Among the key ingredients for the proof arethe following well-known area and co-area formulae. For example, theyare already found in [37], see also [22]. We include a short proof for thesake of convenience. Let `1(X,m) = f : X → R |∑x∈X |f(x)|m(x) <∞.


Lemma 3.4. Let f ∈ `1(X,m), f ≥ 0 and Ωt := x ∈ X | f(x) > t.Then,

1

2

∑

x,y∈Xb(x, y)d(x, y)|f(x)− f(y)| =

∫ ∞

0

|∂Ωt|dt,

where the value ∞ on both sides of the equation is allowed, and

∑

x∈Xf(x)m(x) =

∫ ∞

0

m(Ωt)dt.

Proof. For x, y ∈ X, x ∼ y with f(x) 6= f(y), let the interval Ix,y begiven by Ix,y := [f(x)∧ f(y), f(x)∨ f(y)) and let |Ix,y| = |f(x)− f(y)|be the length of Ix,y. Then, (x, y) ∈ ∂Ωt if and only if t ∈ Ix,y. Hence,by Fubini’s theorem,

∫ ∞

0

|∂Ωt|dt =1

2

∫ ∞

0

∑

x,y∈Xb(x, y)d(x, y)1Ix,y(t)dt

=1

2

∑

x,y∈Xb(x, y)d(x, y)

∫ ∞

0

1Ix,y(t)dt

=1

2

∑

x,y∈Xb(x, y)d(x, y)|f(x)− f(y)|.

Note that x ∈ Ωt if and only if 1(t,∞)(f(x)) = 1. Again, by Fubini’stheorem,

∫ ∞

0

m(Ωt)dt =

∫ ∞

0

∑

x∈Xm(x)1(t,∞)(f(x))dt

=∑

x∈Xm(x)

∫ ∞

0

1(t,∞)(f(x))dt =∑

x∈Xm(x)f(x).

3.3. Form estimates.

Lemma 3.5. For U ⊆ X and u ∈ D with support in U and ‖u‖m = 1

Q(u) ≥ α(U)2

2.

Moreover, if m ≥ n and d ≥ 1 or d ≤ 1 for neighbors, then

Q(u)2 − 2Q(u) + α(U)2 ≤ 0.


Proof. Let u ∈ Cc(X). We calculate, using the co-area formulae abovewith f = u2,

α‖u‖2m = α

∫ ∞

0

m(Ωt)dt ≤∫ ∞

0

|∂Ωt|dt

=1

2

∑

x,y∈Xb(x, y)d(x, y)|u2(x)− u2(y)|

≤ Q(u)12

(1

2

∑

x,y∈Xb(x, y)d(x, y)2(u(x) + u(y))2

) 12

≤ Q(u)12

(2∑

x∈Xu(x)2

∑

y∈Xb(x, y)d(x, y)2

) 12 ≤ 2

12Q(u)

12‖u‖m,

where the final estimate follows from the intrinsic metric property. Thesecond statement follows if we use in the above estimates

1

2

∑

x,y∈Xb(x, y)d(x, y)2(u(x) + u(y))2

= 2∑

x,y∈Xb(x, y)d(x, y)2u(x)2 − 1

2

∑

x,y∈Xb(x, y)d(x, y)2(u(x)− u(y))2

≤ 2‖u‖2m −Q(u),

where we distinguish the cases d ≥ 1 and d ≤ 1: For the first case weuse that d is intrinsic and that −d(x, y)2 ≤ −1. For the second case,we estimate d(x, y) ≤ 1 in the first line and then use n ≤ m. Thestatement follows by the density of Cc(X) in D.

3.4. Proof of the theorems.

Proof of Theorem 3.1 and Theorem 3.2. By virtue of Lemma 3.5, thestatements follow by the variational characterization of λ0 via theRayleigh Ritz quotient: λ0 = infu∈D,‖u‖=1 Q(u).

Proof of Theorem 3.3. Let QU , U ⊆ X, be the restriction of Q to

Cc(U)‖·‖Q

and LU be the corresponding operator. Note that QU = Q

on Cc(U). The assumption Cc(X) ⊆ D(L) clearly implies LCc(X) ⊆`2(X,m) which is equivalent to the fact that functions y 7→ b(x, y)/m(y)are in `2(X,m) for all x ∈ X, see [36, Proposition 3.3]. This impliesthat for any finite set K ⊆ X the operator LX\K is a compact pertur-bation of L. Thus, from Lemma 3.5 we conclude

λess0 (L) = λess

0 (LX\K) ≥ λ0(LX\K) = infu∈Cc(X\K),‖u‖=1

Q(u) ≥ α(X \K)2

2.


This implies the statement of Theorem 3.3.

3.5. Upper bounds for the bottom of spectrum. In this sectionwe show an upper bound for λ0(L) by α as in [9, 11, 12] for uniformlydiscrete metric spaces.

Theorem 3.6. Let d be an intrinsic metric such that (X, d) is uni-formly discrete with lower bound δ > 0. Then, λ0(L) ≤ α/δ.

Proof. By assumption we have d ≥ δ > 0 away from the diagonal.It follows that |∂W | ≥ δ

∑(x,y)∈∂W b(x, y) = δQ(1W ) for all W ⊆ X

finite. By the inequality δλ0(L) ≤ δQ(1W )/‖1W‖2 ≤ |∂W |/m(W ), weconclude the statement.

The example below shows that, in general, there is no upper boundby α only.

Example 3.7. Let b0 : X × X → 0, 1 be a k-regular rooted treewith root x0 ∈ X (that is, each vertex has k forward neighbors).Furthermore, let b1 : X × X → 0, 1 be such that b1(x, y) = 1 ifand only if x and y have the same distance to x0 with respect to thenatural graph distance in b0, and b1(x, y) = 0 otherwise. Now, letb = b0 + b1, m ≡ 1 and let d be given by the path metric with weightsw(x, y) = (n(x) ∨ n(y))−

12 for x ∼ y. Then, α = αd,m = 0 which can

be seen by |∂Br|/m(Br) ≤ k−(r−1)/2 → 0 as r → ∞, where Br is theset of vertices that have distance less or equal r to with respect to thenatural graph metric.

On the other hand, by [39, Theorem 2] the heat kernel pt(x0, ·) ofthe graph b equals the corresponding heat kernel on the k-regular treeb0. Hence, by a Li type theorem, see [24, Theorem 8.1] or [38, Corol-

lary 5.6], we get 1t

log pt(x0, y) → −λ0(L) = −(k + 1 − 2√k) for any

y ∈ X as t → ∞ (see also [39, Corollary 6.7]). As α = 0, this showsthat α can yield no upper bound without further assumptions.

Nevertheless, the example does not exclude the possibility that thereis a different intrinsic metric which might yield a reasonable upperbound. Hence, one might ask whether there exist examples for whichevery intrinsic metric fails to give an upper bound or, otherwise, if onecan always find an intrinsic metric that yields an upper bound.

4. Potentials

In this section we briefly discuss how the strategy proposed in [37] toincorporate potentials into the inequalities can be applied to the new


definition of the Cheeger constant. This yields a Cheeger estimate forall regular Dirichlet forms on discrete sets (cf. [36, Theorem 7]).

Let b be a graph over a discrete measure space (X,m). Furthermore,let c : X → [0,∞) be a potential and define

Qb,c(u) =1

2

∑

x,y∈Xb(x, y)(u(x)− u(y))2 +

∑

x∈Xc(x)u(x)2

on D(Qb,c) = Cc(X)‖·‖Qb,c and let Lb,c be the corresponding operator.

Let (X ′, b′,m′) be a copy of (X, b,m). Let X = X ∪ X ′, m : X →(0,∞) such that m|X = m, m|X′ = m′ and let b : X × X → [0,∞)

be given by b|X×X = b, b|X′×X′ = b′, b(x, x′) = c(x) = c′(x′) for corre-

sponding vertices x ∈ X and x′ ∈ X ′ and b ≡ 0 otherwise. Then, the

restriction Qb,X of the form Qb on `2(X, m) to D(Qb,X) = Cc(X)‖·‖Q

b

satisfies

D(Qb,c) = D(Qb,X) and Qb,c = Qb,X .

Let d : X ×X → [0,∞) be an intrinsic metric for b over (X,m) andassume there is a function δ : X → [0,∞) such that

∑

y∈Xb(x, y)d(x, y)2 + c(x)δ(x)2 ≤ m(x) for all x ∈ X.

Example 4.1. (1) For a given intrinsic metric d a possible choice

for the function δ is δ(x) = ((m(x)−∑y∈X b(x, y)d(x, y)2)/c(x))12 if

c(x) > 0 and 0 otherwise.(2) Choose d as the path metric induced by the edge weights w(x, y) =

(( mn+c

)(x) ∧ ( mn+c

)(y))12 for x ∼ y and δ as in (1). If c > 0, then δ > 0.

We next define d. Since we are only interested in the subgraph Xof X, we do not need to extend d to all of X but only set d|X×X = d

and d(x, x′) = δ(x) for the corresponding vertex x′ ∈ X ′ of x. Definingα(X) = αd,m(X) by

α(X) = infW⊆Xfinite

|∂W |dm(W )

with |∂W |d =∑

(x,y)∈∂W b(x, y)d(x, y) +∑

x∈W c(x)δ(x) implies that

α(X) = αd,m(X), where the right hand side is the Cheeger constant of

the subgraph X ⊆ X as in Section 2.3. Hence, we get

λ0(Lb,c) ≥α(X)2

2by Lemma 3.5 and the arguments from the proof of Theorem 3.1.


5. Upper bounds by volume growth

In this section we relate the isoperimetric constant to the exponentialvolume growth of the graph. Let b be a graph over a discrete measurespace (X,m) and let d be an intrinsic metric. We let Br(x) = y ∈ X |d(x, y) ≤ r and define the exponential volume growth µ = µd,m by

µ = lim infr→∞

infx∈X

1

rlog

m(Br(x))

m(B1(x)).

Other than for the classical notions of isoperimetric constants and ex-ponential volume growth on graphs (see [5, 11, 18, 43]), it is, geometri-cally, not obvious that α = αd,m and µ = µd,m can be related. However,given a Brooks-type theorem, the proof is rather immediate. Therefore,let the maximal form domain be given by

Dmax := u ∈ `2(X,m) | Qmax(u) =1

2

∑

x,y∈Xb(x, y)(u(x)− u(y))2 <∞.

Theorem 5.1. If D = Dmax, then 2α ≤ µ. In particular, this holds ifone of the following assumptions is satisfied:

(a) The graph b is locally finite and d is an intrinsic path metricsuch that (X, d) is metrically complete.

(b) Every infinite path of vertices has infinite measure.

Proof. Under the assumption D = Dmax we have λ0(L) ≤ µ2/8 by [25,Theorem 4.1]. (Note that the 8 in the denominator as opposed to the4 found in [25] is explained in [25, Remark 3].) Thus, the statementfollows by Theorem 3.1. Note that, by [32, Theorem 2] and [24, Corol-lary 6.3], (a) implies D = Dmax and, by [36, Theorem 6], (b) impliesD = Dmax.

6. Lower bounds by curvature

In this section we give a lower bound on the isoperimetric constantby a quantity that is sometimes interpreted as curvature [11, 29, 39].Let b be a graph over (X,m) and let d be an intrinsic metric.

6.1. The lower bound. We fix an orientation on a subset of the edges,that is, we choose E+, E− ⊂ X × X with E+ ∩ E− = ∅ such that(x, y) ∈ E+ if and only if (y, x) ∈ E−. We define the curvature withrespect to this orientation by K : X → R

K(x) =1

m(x)

( ∑

(x,y)∈E−

b(x, y)d(x, y)−∑

(x,y)∈E+

b(x, y)d(x, y))

Let us give an example for a choice of E±.


Example 6.1. Let b take values in 0, 1, m be the vertex degreefunction n, and d be the natural graph metric. For some fixed vertexx0 ∈ X, let Sr be the spheres with respect to d around x0 and |x| = rfor x ∈ Sr. We choose E± such that outward (inward) oriented edgesare in E+ (E−), i.e., (x, y) ∈ E+, (y, x) ∈ E− if x ∈ Sr−1, y ∈ Srfor some r and x ∼ y. Then K(x) = (n−(x) − n+(x))/n(x), wheren±(x) = #y ∈ S|x|±1 | y ∼ x and #A denotes the cardinality of A.

The following theorem is an analogue to [11, Lemma 1.15] and [13,Proposition 3.3] which was also used in [49, 50] to estimate the bottomof the essential spectrum.

Theorem 6.2. If −K ≥ k ≥ 0, then α ≥ k.

Proof. Let W be a finite set and denote by 1W the corresponding char-acteristic function. Furthermore, let σ(x, y) = ±d(x, y) for (x, y) ∈ E±and zero otherwise. We calculate directly

km(W ) ≤ −∑

x∈WK(x)m(x) =

∑

x∈X1W (x)

∑

y∈Xb(x, y)σ(x, y)

=1

2

(∑

x∈X1W (x)

∑

y∈Xb(x, y)σ(x, y)−

∑

y∈X1W (y)

∑

x∈Xb(x, y)σ(x, y)

)

≤ 1

2

∑

x,y∈Xb(x, y)d(x, y)|1W (x)− 1W (y)| = |∂W |,

where we used∑

y b(x, y)d(x, y) < ∞ and the antisymmetry of σ inthe second step. This finishes the proof.

6.2. Example of antitrees. In the final subsection we give an exam-ple of an antitree for which Theorem 6.2 together with Theorem 3.1yields a better estimate than the estimates known before. Recently,antitrees received some attention as they provide examples of graphsof polynomial volume growth (with respect to the natural graph met-ric) that are stochastically incomplete and have a spectral gap, see[4, 23, 25, 31, 39, 51].

For a given graph b : X×X → 0, 1 with root x0 ∈ X and measurem ≡ 1, let Sr be the vertices that have natural graph distance r to x0

as above. We call a graph an antitree if every vertex in Sr is connectedto all vertices in Sr+1 ∪ Sr−1 and to none in Sr.

In [25], it is shown that λ0(L) = 0 whenever limr→∞ log #Sr/ log r <2. It remains open by this result what happens in the case of an antitreewith #Sr−1 = r2. The classical Cheeger constant αclassical = α1,n forthe normalized Laplacian with the natural graph metric which is given


as the infimum over #∂W/n(W ) (with W ⊆ X finite) is zero. Thiscan be easily checked by choosing distance balls Br =

⋃rj=0 Sj as test

sets W . Hence, the estimate λ0(L) ≥ (1 −√

1− α2) infx∈X n(x) withα = α1,n found in [34] is trivial.

Likewise, the estimates presented in [39] and [50], which uses anunweighted curvature, also give zero as a lower bound for the bottomof the spectrum in this case.

By Theorem 6.2 we obtain a positive estimate for the Cheeger con-stant α = αd,1 with the path metric induced by the edge weights

w(x, y) = (n(x)∨n(y))−12 , x ∼ y for the antitree with #Sr−1 = r2. We

pick E± as in Example 6.1 above and obtain a positive lower bound for−K. In particular, Theorem 6.2 shows that α > 0 and, thus, λ0(L) > 0by Theorem 3.1 for the antitree satisfying #Sr−1 = r2.

Acknowledgements. F.B. thanks Jurgen Jost for introducing him to the topic and formany stimulating discussions during the last years. The research leading to these resultshas received funding from the European Research Council under the European Union’sSeventh Framework Programme (FP7/2007-2013) / ERC grant agreement n 267087 andthe Alexander von Humboldt foundation.

M.K. thanks Daniel Lenz for sharing generously his knowledge about intrinsic metricsand he enjoyed vivid discussions with Markus Seidel, Fabian Schwarzenberger and MartinTautenhahn. M.K. also gratefully acknowledges the financial support from the GermanResearch Foundation (DFG).

R.K. thanks Jozef Dodziuk for introducing him to Cheeger constants and for numerousinspiring discussions. Support for this project was provided by PSC-CUNY Awards,jointly funded by the Professional Staff Congress and the City University of New York.

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[49] R. K. Wojciechowski, Stochastic completeness of graphs, ProQuest LLC, AnnArbor, MI, 2008. Thesis (Ph.D.)–City University of New York.

[50] R. K. Wojciechowski, Heat kernel and essential spectrum of infinite graphs,Indiana Univ. Math. J. 58 (2009), no. 3, 1419–1441.


[51] R. K. Wojciechowski, Stochastically incomplete manifolds and graphs, Randomwalks, boundaries and spectra, Progr. Prob., vol. 64, Birkhauser Verlag, Basel,2011, 163–179.

Frank Bauer, Department of Mathematics, Harvard University, Cam-bridge, MA 02138, USA and Max Planck Institute for Mathematics inthe Sciences, Inselstrasse 22, 04103 Leipzig, Germany


Matthias Keller, Einstein Institute of Mathematics, The HebrewUniversity of Jerusalem, Jerusalem 91904, Israel and Mathematis-ches Institut Friedrich-Schiller-Universitat Jena, Ernst-Abbe-Platz2, 07743 Jena, Germany


Rados law K. Wojciechowski, York College of the City Universityof New York, Jamaica, NY 11451, USA


CHAPTER 11

S. Haeseler, M. Keller, R. Wojciechowski, Volumegrowth and bounds for the essential spectrum for

Dirichlet forms, Journal of the LondonMathematical Society 88 (2013), 883–898.

255

J. London Math. Soc. (2) 88 (2013) 883–898 C2013 London Mathematical Societydoi:10.1112/jlms/jdt029

Volume growth and bounds for the essential spectrum forDirichlet forms

Sebastian Haeseler, Matthias Keller and Radoslaw K. Wojciechowski

Abstract

We consider operators arising from regular Dirichlet forms with vanishing killing term. We givebounds for the bottom of the (essential) spectrum in terms of exponential volume growth withrespect to an intrinsic metric. As special cases, we discuss operators on graphs. When the volumegrowth is measured in the natural graph distance (which is not an intrinsic metric), we discussthe threshold for positivity of the bottom of the spectrum and finiteness of the bottom of theessential spectrum of the (unbounded) graph Laplacian. This threshold is shown to lie at cubicpolynomial growth.

1. Introduction and main results

In 1981, Brooks proved that the bottom of the essential spectrum of the Laplace Beltramioperator on a complete non-compact Riemannian manifold with infinite measure can bebounded by the exponential volume growth rate of the manifold [2]. Following this, similarresults were proved in various contexts; see [3, 6, 12, 13, 22–25, 28]. Very recently, it wasshown in [19] that such a result fails to be true in the case of the (non-normalized) graphLaplacian when the volume is measured with respect to the natural graph distance. Indeed,there are graphs of cubic polynomial volume growth that have positive bottom of the spectrumand slightly more than cubic growth already allows for a purely discrete spectrum. This suggeststhat one should look at other candidates for a metric on a graph.

In this paper, we work in the context of regular Dirichlet forms (without killing term) and usethe corresponding concept of an intrinsic metric (see [5, 28]) to prove a Brooks-type theorem.The purpose of this approach is threefold. First, we provide a setup which includes all knownexamples (and various others, for example, quantum graphs) and give a unified treatment.Additionally, our estimates are better than most of the previous results (by replacing a lim supby a lim inf for the essential spectrum). Second, our method of proof seems to be much clearerand simpler than most of the previous works. Finally, graph Laplacians are now included andthe disparity discussed above is resolved by considering suitable metrics. As an application, wecan now prove that the examples found in [19] for Laplacians on graphs do indeed give theborderline for positive bottom of the spectrum. In particular, for the natural graph distance,the threshold for zero bottom of the essential spectrum and the discreteness of the spectrumlies at cubic growth.

Let X be a locally compact separable metric space and m be a positive Radon measure offull support. Let E be a closed, symmetric, non-negative form on the Hilbert space L2(X,m) ofreal-valued square-integrable functions with domain D. We assume that E is a regular Dirichlet

Received 3 July 2012; revised 9 January 2013; published online 26 September 2013.

2010 Mathematics Subject Classification 47D07 (primary), 58J50, 35P05, 39A12 (secondary).

The research of Radoslaw K. Wojciechowski was partially sponsored by the Fundacao para a Ciencia ea Tecnologia through project PTDC/MAT/101007/2008 and by PSC-CUNY Awards, jointly funded by theProfessional Staff Congress and the City University of New York. Matthias Keller appreciates the financialsupport of the German Science Foundation (DFG).

884 S. HAESELER, M. KELLER AND R. K. WOJCIECHOWSKI

form without killing term (for background on Dirichlet forms see [7]; more details are given inSubsection 2.1). Let L be the positive self-adjoint operator arising from E . Define

λ0(L) := inf σ(L) and λess0 (L) := inf σess(L),

where σess(L) denotes the essential spectrum of L.We let ρ be an intrinsic metric in the sense of [5]. (Note that an intrinsic metric is, in general,

a pseudometric.) For x0 ∈ X and r 0, we define the distance ball Br = Br(x0) = x ∈ X |ρ(x, x0) r. Let the exponential volume growth be defined as

μ = lim infr→∞

1

rlogm(Br(x0)).

Note that, in contrast to previous works on manifolds [2] and graphs [6], we consider a lim infhere, rather than a lim sup.

If ρ takes values in [0,∞), then X =⋃

r Br(x0). In this case, μ does not depend on theparticular choice of x0. There is another constant first introduced in [28] which we call theminimal exponential volume growth and which is defined as

μ = lim infr→∞

1

rinfx∈X

logm(Br(x))

m(B1(x)).

In this paper, we prove the following theorem.

Theorem 1.1. Let L be the positive self-adjoint operator arising from a regular Dirichletform E without killing term and let ρ be an intrinsic metric such that all distance balls arecompact. Then,

λ0(L) μ2

4.

If, additionally, m(⋃

r Br(x0)) = ∞ for some x0, then

λess0 (L) μ2

4.

This has the following immediate corollary. The corollary has various consequences, forexample, the exponential instability of the semigroup (e−tL)t0 on Lp(X,m), p ∈ [1,∞];see [28, Corollary 2].

Corollary 1.2. Suppose that (X, ρ) is of subexponential growth, that is, μ = 0(respectively, μ = 0). Then, λ0(L) = 0 (respectively, λess0 (L) = 0).

Remark 1.3. (a) Let us discuss Theorem 1.1 in the perspective of the present literature:for the Laplace Beltrami operator on a Riemannian manifold an estimate for λess0 can be foundin [2]; see also [12]. For strongly local Dirichlet forms, the statement for λ0 is proved in [28] andfor λess0 in [24]; see also [22, 23] for more subtle estimates involving λ0 in the case of manifolds.For non-local operators, such results were known only for normalized Laplacians on graphs;see [3, 6, 13, 25]. These operators are of a very special form, in particular, they are alwaysbounded. For unbounded Laplacians on graphs, the conclusions of the theorem do not hold ifone considers volume with respect to the natural graph metric; see [19]. However, by Frank,Lenz and Wingert [5] (see also [9]), there is now a suitable notion of an intrinsic metric fornon-local forms. Let us stress that our result covers the results in [2, 3, 6, 25, 28]. Results of the

VOLUME GROWTH AND BOUNDS FOR THE ESSENTIAL SPECTRUM 885

type found in [12, 13] could certainly also be obtained with slightly more technical effort whichwe avoid here for the sake of the clarity of presentation.

(b) Despite the fact that our result is much more general, we have a unified method of prooffor the bounds on the spectrum and the essential spectrum. For the essential spectrum, theproof is significantly simpler than the one of [2, 6] as we avoid a cutoff procedure by using testfunctions which converge weakly to zero.

(c) Indeed, we prove a slightly more general result than above for non-local forms inSubsection 3.2. In particular, for some special cases we prove much better estimates and recoverthe results of [3, 6, 25] in Corollary 4.6 in Subsection 4.1.

(d) If we assume that ρ takes values in [0,∞), then we can clearly replace the assumptionthat m(

⋃r Br(x0)) = ∞ with m(X) = ∞. The case when m(X) < ∞ is notably different; see

[8] for more details.(e) If infx∈X m(B1(x)) > 0, then one can also show λess0 (L) μ2/4; see Remark 2.3 in

Subsection 2.3.(f) Our result deals exclusively with Dirichlet forms with vanishing killing term. The major

challenge in the case of a non-vanishing killing term is to give a proper definition of volumewhich incorporates the killing term. We shortly discuss a strategy for approaching this case: weneed a positive generalized harmonic function u, that is, E(u, ϕ) = 0 for all ϕ ∈ D, where u isassumed to be locally in the domain of E (this space is introduced in [5] as D∗

loc). Such a functionexists in many settings (see, for example, [3, 11, 21]); the result guaranteeing the existenceof such a function is often referred to as an Allegretto–Piepenbrink-type theorem. Then, bya ground state representation (see [5, Theorem 10.1]), one obtains a form Eu with vanishingkilling term such that E = Eu on the intersection of their domains. Now we can apply themethods above for Eu to derive the result for E . However, as shown in [11], there are examplesof non-locally finite weighted graphs that do not have such a generalized harmonic function.Therefore, it would be interesting to find sufficient conditions under which the approach abovecould be carried out.

Let us highlight one of the applications of our results for graphs. Let Δ be the graph Laplacianon 2(X) acting as

Δϕ(x) =∑

y∼x

(ϕ(x) − ϕ(y))

(for more details, see Subsections 4.1 and 4.2). Moreover, let Bdr , for r 0, be balls with respect

to the natural graph distance d defined as the smallest number of edges in a path between twovertices. It has to be stressed that this metric is not an intrinsic metric for Δ. However, we willshow in Theorem 4.7 that, if the growth of the balls Bd

r is smaller than r3−ε for any ε > 0, thenλ0(Δ) = λess0 (Δ) = 0 and, if it is less than r3, then λess0 (Δ) < ∞. We demonstrate by examplesthat this result is sharp; see Subsection 4.2.

The paper is structured as follows. In Section 2, we recall some basic facts about Dirichletforms and intrinsic metrics. Moreover, we give a bound on the bottom of the essential spectrumvia weak null sequences and introduce the test functions. In Section 3, we prove the crucialestimate for the strongly local and the non-local parts of the Dirichlet form and prove the maintheorem. In Section 4, we discuss the result for weighted graphs and prove the polynomialgrowth bound discussed above.

Note added: after this work was completed, we learned about the recent preprint of MatthewFolz [4] which contains related material in the special case of locally finite graphs. In particular,the main result [4, Theorem 1.3] is a special case of Corollary 4.4 and [4, Theorem 1.4] givesslightly better estimates than Corollaries 4.4 and 4.6.


2. Preliminaries

In this section, we introduce the basic notions and concepts. The first subsection is devoted torecalling the setting of Dirichlet forms. In the second subsection, we prove an estimate for thebottom of the essential spectrum and in the third subsection we discuss the basic propertiesof the test functions that are used to prove our result.

2.1. Dirichlet forms

In this section, we recall some elementary facts about Dirichlet forms; see, for example, [7]and, for recent work on non-local forms [5].

As above, let X be a locally compact separable metric space and let m be a positive Radonmeasure of full support. We consider all functions on X to be real valued, but, by complexifyingthe corresponding Hilbert spaces and forms, we could also consider complex-valued functions. Aclosed non-negative form on L2(X,m) consists of a dense subspace D ⊆ L2(X,m) and a sesqui-linear non-negative map E : D ×D → R such that D is complete with respect to the form norm‖ · ‖E =

√E(·, ·) + ‖ · ‖2, where ‖ · ‖ always denotes the L2 norm. We write E(u) := E(u, u) for

u ∈ D.A closed non-negative form (E ,D) is called a Dirichlet form if, for any u ∈ D and any

normal contraction c : R → R, we have c u ∈ D and E(c u) E(u). Here, c is a normalcontraction if c(0) = 0 and |c(x) − c(y)| |x− y| for x, y ∈ R. A Dirichlet form is called regularif D ∩ Cc(X) is dense both in (D, ‖ · ‖E) and (Cc(X), ‖ · ‖∞), where Cc(X) is the space ofcontinuous compactly supported functions.

A function f : X → R is said to be quasi-continuous if, for every ε > 0, there is an open setU ⊆ X with

cap(U) := inf‖v‖E | v ∈ D, 1U v ε,

such that f |X\U is continuous (where inf ∅ = ∞ and 1U is the characteristic function of U).For a regular Dirichlet form (E ,D), every u ∈ D admits a quasi-continuous representative; see[7, Theorem 2.1.3]. In the following, we assume that, when considering u as a function, wealways choose a quasi-continuous representative.

There is a fundamental representation theorem for regular Dirichlet forms called theBeurling–Deny formula; see [7, Theorems 3.2.1 and 5.2.1]. It states that

E(u) =

∫

X

dΓ(c)(u) +

∫

X×X\d(u(x) − u(y))2 dJ(x, y) +

∫

X

u(x)2 dk(x),

where we choose a quasi-continuous representative of u in the second and third integrals. Here,Γ(c) is a positive semidefinite bilinear form on D ×D, with values in the signed Radon measureson X, which is strongly local, that is, satisfies Γ(c)(u, v) = 0 if u is constant on the support ofv. Secondly, J is a non-negative Radon measure on X ×X \ d (which is X ×X without thediagonal d := (x, x) | x ∈ X) and, finally, k is a non-negative Radon measure on X. The firstterm on the right-hand side is called the strongly local part of E , the second term is called thejump part and the third term is called the killing term. The measure J gives rise to a formΓ(j) on D ×D with values in the signed Radon measures on X (where the j refers to ‘jump’)which is characterized by

∫

K

dΓ(j)(u) =

∫

K×X\d(u(x) − u(y))2 dJ(x, y),

for K ⊆ X compact and u ∈ D. The focus of this paper is on regular Dirichlet forms E withoutkilling term, that is, k ≡ 0. Thus, we define

Γ := Γ(c) + Γ(j).


The space D∗loc of functions locally in the domain of E was introduced in [5] and is important

for the definition of intrinsic metrics. It is defined as the set of functions u ∈ L2loc(X,m) such

that for all open and relatively compact sets G there is a function v ∈ D such that u and vagree on G and for all compact K ⊆ X,∫

K×X\d(u(x) − u(y))2 dJ(x, y) < ∞.

We can extend Γ(c) and Γ(j) to D∗loc; see [7, Remarks after the proof of Theorem 3.2.2] and

[5, Proposition 3.3].For the strongly local part, we have a chain rule (see [7, Theorem 3.2.2]) as follows: for

ϕ : R → R continuously differentiable with bounded derivative ϕ′,

Γ(c)(ϕ(u), v) = ϕ′(u)Γ(c)(u, v), u, v ∈ D∗loc ∩ L∞(X,m).

A pseudometric is a map ρ : X ×X → [0,∞] which is symmetric, satisfies the triangleinequality and ρ(x, x) = 0 for all x ∈ X. For A ⊆ X, we define the map ρA : X → [0,∞] by

ρA(x) = infy∈A

ρ(x, y).

If ρ is a pseudometric and T > 0, then ρ ∧ T is a pseudometric and we have (ρ ∧ T )A = ρA ∧ Tand |ρA(x) ∧ T − ρA(y) ∧ T | ρ(x, y).

By Frank, Lenz and Wingert [5, Definition 4.1], a pseudometric ρ is called an intrinsic metricfor the Dirichlet form E if there are Radon measures m(c) and m(j) with m(c) +m(j) m suchthat for all A ⊆ X and all T > 0 the functions ρA ∧ T are in D∗

loc ∩ C(X), where C(X) denotesthe set of continuous functions on X, and satisfy

Γ(c)(ρA ∧ T ) m(c) and Γ(j)(ρA ∧ T ) m(j).

This implies that if A ⊆ X is such that ρA(x) < ∞ for all x ∈ X, then ρA ∈ D∗loc ∩ C(X) and

Γ(ρA) m (see [5, Proposition 4.4]).

2.2. An estimate for the bottom of the essential spectrum

The following Persson-type theorem seems to be standard in certain settings; see [10, 26].However, since we are not able to find a proper reference in the literature which covers ourcase, we include a short proof.

Proposition 2.1. Let h be a closed quadratic form on L2(X,m) that is bounded frombelow and let H be the corresponding self-adjoint operator. Assume that there is a normalizedsequence (fn) in D(h) that converges weakly to zero in L2(X,m). Then,

λess0 (H) lim infn→∞

h(fn).

Proof. Without loss of generality, assume h 0 and λess0 (H) > 0. Let 0 < λ < λess0 (H).We will show that there is an N 0 such that h(fn) > λ for all n N . Let λ1 be such thatλ < λ1 < λess0 (H) and let ε > 0 be arbitrary. As λ1 < λess0 (H), the spectral projection E(−∞,λ1]

of H and the interval (−∞, λ1] is a finite rank operator. Therefore, as (fn) converges weaklyto zero, there is an N 0 such that ‖E(−∞,λ1]fn‖2 < ε for n N . Letting νn be the spectralmeasure of H with respect to fn, we estimate for n N ,

h(fn) ∫∞

λ1

t dνn(t) λ1

∫∞

λ1

dνn(t) = λ1(‖fn‖2 − ‖E(−∞,λ1]fn‖2) > λ1(1 − ε),

where we used λ1 0 as h 0. We conclude the asserted inequality by choosing ε = (λ1 −λ)/λ1 > 0.


2.3. The test functions

In this section, we introduce the sequence of test functions that we will use to estimate thebottom of the (essential) spectrum.

For r ∈ N, x0 ∈ X,α > 0, define

fr,x0,α : X → [0,∞), x → ((eαr ∧ eα(2r−ρ(x0,x))) − 1) ∨ 0.

Then, for fixed r, α, x0, we have f |Br≡ eαr − 1, f |B2r\Br

= eα(2r−ρ(x0,·)) − 1 and f |X\B2r≡ 0.

Clearly, f is spherically homogeneous, that is, there exists h : [0,∞) → [0,∞) such that f(x) =h(ρ(x0, x)). The definition of f combines ideas from [2, 6, 28].

Moreover, for r ∈ N, x0 ∈ X, α > 0, let gr,x0,α : X → [0,∞) be given by

gr,x0,α = (fr,x0,α + 2)1B2r.

Lemma 2.2. Let x0 ∈ X, fr = fr,x0,α and gr = gr,x0,α for r 0, α > 0. If μ < ∞, then wehave the following conditions:

(a) fr, gr ∈ L2(X,m) for all r 0, α > 0;(b) if m(

⋃r Br) = ∞, then fr/‖fr‖ converges weakly to 0 as r → ∞;

(c) if α > μ/2, then there is a sequence (rk) such that ‖grk‖/‖frk‖ → 1 as k → ∞.

In general (possibly μ = ∞),

(d) if α > μ/2, then there are sequences (xk) in X and (rk) such that fk = frk,xk,α, gk =grk,xk,α ∈ L2(X,m) and we have that ‖gk‖/‖fk‖ → 1 as k → ∞.

Proof. (a) Since we assume μ < ∞, it follows that m(Br(x0)) < ∞ for all r 0. Therefore,fr, gr ∈ L2(X,m) for all r 0 since fr and gr are bounded and supported in B2r.

(b) Let ψ ∈ L2(X,m) with ‖ψ‖ = 1, ε > 0 and set ϕ = ψ1⋃Br. There exists R > 0 such that

‖ϕ1X\BR‖ ε/2. Moreover, let r R be such that m(BR) ε2m(Br)/4 (this choice is possible

since m(⋃Br) = ∞). We conclude by the Cauchy–Schwarz inequality and ‖fr1BR

‖ ε/2‖fr‖that

〈ϕ, fr〉 = 〈ϕ1BR, fr〉 + 〈ϕ1X\BR

, fr〉 ‖ϕ‖‖fr1BR‖ + ‖ϕ1X\BR

‖‖fr‖ ε‖fr‖.

As supp fr ⊆ ⋃sBs, it follows that 〈ψ, fr〉 = 〈ϕ, fr〉 for r 0 which proves (b).

Before we prove (c), we show (d) and indicate how to adapt the proof to (c) afterwards. Ifμ = ∞, then there is nothing to prove so assume μ < ∞. Let 0 < ε < α− μ/2. By the definitionof μ, there are sequences (rk) of increasing positive numbers and (xk) of elements in X suchthat

m(B2rk(xk))

m(B1(xk)) e(2μ+ε)rk , k 0.

We set fk = frk,xk,α, gk = grk,xk,α. Since m(B2rk(xk)) < ∞ and the functions fk, gk aresupported in B2rk(xk) and bounded, they are in L2(X,m). By definition gk = gk1B2rk

=

(fk + 2)1B2rk, k 0. Using the inequalities (a+ b)2 (1/(1 − ε))a2 + (1/ε)b2 and ‖fk‖2

m(Brk(xk))(eαrk − 1)2 m(Brk(xk)) e2αrk/c for c = (1 − e−αr0)−2 > 0, we get

‖gk‖2‖fk‖2

(‖fk‖ + 2√m(B2rk(xk)))2

‖fk‖2 (1/(1 − ε))‖fk‖2 + (4/ε)m(B2rk(xk))

‖fk‖2

1

(1 − ε)+

4c

ε

m(B2rk(xk))

m(Brk(xk))e−2αrk .


For rk large enough, we have by definition of μ,

m(Brk(xk))

m(B1(xk)) inf

x∈X

m(Brk(x))

m(B1(x)) e(μ−ε)rk .

Thus, by the choice of (rk) and (xk), we have m(B2rk)/m(Brk) e(μ+2ε)rk . As 0 < ε < α−μ/2,

‖gk‖2‖fk‖2

1

(1 − ε)+

4c

εe(μ+2ε−2α)rk → 1

(1 − ε)as k → ∞.

Since ε can be chosen arbitrarily small and ‖gk‖ ‖fk‖, we deduce the statement.For (c), we choose (xk) to be x0 and follow the lines of the proof replacing μ by μ.

Remark 2.3. If infx∈X m(B1(x)) > 0, then fk/‖fk‖ of (d) also converges weakly to zeroas k → ∞.

The following auxiliary estimates will later give us bounds for the Lipshitz constants of fr,x,α.

Lemma 2.4. Let α > 0. For all R 0, one has

(eαR − 1)2

(e2αR + 1) α2R2

2.

Moreover, for R ∈ [0, 1] one has

(eαR − 1)2

(e2αR + 1) R2(eα − 1)

2

(R2e2α + 1).

Proof. For the first statement, let s = αR and check via a series expansion thats → s2(e2s + 1) − 2(es − 1)

2is non-negative. The second statement follows by eαR − 1

R(eα − 1) for R ∈ [0, 1] from the series expansion, the elementary inequality a2/((a+ 1)2 +1) b2/((b+ 1)2 + 1) for 0 a b, and 1 −R 0.

Lemma 2.5. Let r ∈ N, x0 ∈ X, α > 0 and set f := fr,x0,α, g := gr,x0,α. Then, for x, y ∈ X,

(f(x) − f(y))2 c(α)(g(x)2 + g(y)2)ρ(x, y)2,

where c(α) = α2/2. If, additionally, ρ(x, y) 1, then c(α) can be chosen to be c(α, ρ(x, y)) =(eα − 1)2/ρ(x, y)2e2α + 1. In particular, f is Lipshitz continuous with Lipshitz constantα(eαr + 1).

Proof. We fix r, α and x0 for the proof. Let x, y ∈ X be given and let s =ρ(x0, x) and t = ρ(x0, y). We define Ds,t := (f(x) − f(y))2. Moreover, we estimate F (R) :=(eαR − 1)2/e2αR + 1, R 0 by c(α)R2 (and by c(α,R)R2 for R 1), from Lemma 2.4. Bysymmetry, we may assume, without loss of generality, that s t so that we have six cases tocheck.

Case 1: If s t r, then Ds,t = 0.Case 2: If s r t 2r, then since t− r t− s = ρ(x0, y) − ρ(x0, x) ρ(x, y) and g(x) =

eαr + 1, g(y) = eα(2r−t) + 1,

Ds,t = (eαr − eα(2r−t))2 = (e2αr + e2α(2r−t))F (t− r) (e2αr + e2α(2r−t))c(α)(t− r)2

c(α)(g(x)2 + g(y)2)ρ(x, y)2.


Case 3: If s r 2r t, then since r t− s ρ(x, y), g(x) = eαr + 1 and g(y) = 0,

Ds,t = (eαr − 1)2 = (e2αr + 1)F (r) (e2αr + 1)c(α)r2 c(α)(g(x)2 + g(y)2)ρ(x, y)2.

Case 4: If r s t 2r, then since t− s ρ(x, y) and g(x) = eα(2r−s) + 1, g(y) = eα(2r−t) + 1,

Ds,t = (eα(2r−s) − eα(2r−t))2 = (e2α(2r−s) + e2α(2r−t))F (t− s)

c(α)(g(x)2 + g(y)2)ρ(x, y)2.

Case 5: If r s 2r t, then since 2r − s t− s ρ(x, y), g(x) = eα(2r−s) + 1 and g(y) = 0,

Ds,t = (eα(2r−s) − 1)2 = (e2α(2r−s) + 1)F (2r − s) c(α)(g(x)2 + g(y)2)ρ(x, y)2.

Case 6: If 2r s t, then Ds,t = 0.The Lipshitz bound follows since g is bounded by eαr + 1.

Lemma 2.6. Let (E ,D) be a regular Dirichlet form and let ρ be an intrinsic metric. Forall r > 0, x0 ∈ X and α > 0, we have fr,x0,α ∈ D∗

loc. Moreover, if B2r(x0) is compact, thenfr,x0,α ∈ D.

Proof. By Lemma 2.5, the functions f := fr,x0,α are Lipshitz continuous for all r > 0, x0and α > 0. Thus, by a Rademacher-type theorem (see [5, Theorem 4.8] or, for strongly localforms, see [27, Theorem 5.1]), we have f ∈ D∗

loc and Γ(f) m. If B2r(x0) is compact, then fis compactly supported which implies f ∈ D (see [5, Theorem 3.4]).

3. Proof of the main theorem

3.1. The strongly local estimate

In this subsection, we give an estimate which will be used to prove the theorem for the stronglylocal part of the Dirichlet form. For given r ∈ N, x0 ∈ X and α > 0, we define f := fr,x0,α andg := gr,x0,α.

Lemma 3.1. Let ρ be an intrinsic metric for a regular strongly local Dirichlet form E . Then,for all r > 0, x0 ∈ X and α > 0 such that f ∈ D, we have

E(f) α2

∫

X

g2 dm(c).

Proof. As E is strongly local, by the chain rule and the fact that ρ is an intrinsic metric weget that

E(f) =

∫

B2r\Br

dΓ(c)(f) =

∫

B2r\Br

dΓ(c)(eα(2r−ρ(x0,·)) − 1)

= α2

∫

B2r\Br

e2α(2r−ρ(x0,·)) dΓ(c)(ρ(x0, ·))

α2

∫

B2r\Br

e2α(2r−ρ(x0,·)) dm(c) α2

∫

X

g2r,x0,α dm(c).

3.2. The non-local estimate

Next, we treat the non-local case. With applications to graphs in the next section in mind, wedo not assume that the jump part is a regular Dirichlet form for now.


For this subsection, let m be a Radon measure on X and let J be a symmetric Radon measureon X ×X \ d such that for every m-measurable A ⊆ X, the set A×X \ d is J measurable andvice versa. Let ρ be a pseudometric on X which is J measurable and assume that for allmeasurable A ⊆ X, ∫

A×X\dρ(x, y)2 dJ(x, y) m(A), (♣)

which immediately implies that for all measurable functions ϕ,∫

X×X\dϕ(x)2ρ(x, y)2 dJ(x, y)

∫

X

ϕ2 dm.

We say that the pseudometric ρ has jump size in [a, b], 0 a b, if for the set Aa,b := (x, y) ∈X ×X | ρ(x, y) ∈ [a, b] \ d,

∫

X×X\dϕ(x, y) dJ(x, y) =

∫

Aa,b

ϕ(x, y) dJ(x, y),

for any positive measurable function ϕ.For given r ∈ N, x0 ∈ X and α > 0, we define f := fr,x0,α and g := gr,x0,α.

Lemma 3.2. Assume that ρ satisfies (♣). For all r ∈ N, x0 ∈ X and α > 0,∫

X×X\d(f(x) − f(y))2 dJ(x, y) 2c(α)

∫

X

g2 dm,

where c(α) = α2/2. If ρ has jump size in [δ, 1] for some 0 δ 1, then c(α) can be chosen tobe c(α, δ) = (eα − 1)2/1 + δ2e2α.

Proof. By Lemma 2.5 and since ρ satisfies (♣),∫

X×X\d(f(x) − f(y))2 dJ(x, y) α2

∫

X×X\dg(x)2ρ(x, y)2 dJ(x, y) α2

∫

X

g2 dm.

Let δ > 0. If ρ has jump size in [δ, 1], then∫

X×X\d(f(x) − f(y))2 dJ(x, y) =

∫

Aδ,1

(f(x) − f(y))2 dJ(x, y)

∫

Aδ,1

g(x)22(eα − 1)2

(1 + ρ(x, y)2 e2α)ρ(x, y)2 dJ(x, y)

2(eα − 1)2

(1 + δ2 e2α)

∫

X×X\dg(x)2ρ(x, y)2 dJ(x, y)

2(eα − 1)2

(1 + δ2 e2α)

∫

X

g2 dm.

3.3. Proof of Theorem 1.1

We now have all of the ingredients to prove our main result.

Proof of Theorem 1.1. By Frank, Lenz and Wingert [5, Lemma 4.7], an intrinsic metricsatisfies (♣). Moreover, under the assumption that the distance balls are compact we havefr,x,α ∈ D for all r > 0, x ∈ X, α > 0 by Lemma 2.6.


Let α > μ/2. By Lemma 2.2(d), there are sequences (xk) and (rk) such that for fk = frk,xk,α,gk = grk,xk,α,

λ0(L) lim infk→∞

E(fk)

‖fk‖2 α2 lim

k→∞‖gk‖2‖fk‖2

= α2,

where the second inequality follows from Lemmas 3.1 and 3.2 and the equality follows fromLemma 2.2(d). Hence, λ0(L) μ2/4.

Let now α > μ/2 and let (rk) be the sequence given by Lemma 2.2(c) for some fixed x0 ∈ Xand let xk = x0 for all k 0. By Lemma 2.2(b), the sequence (fk/‖fk‖) converges weakly tozero in L2(X,m) and, therefore, by Proposition 2.1, Lemmas 3.1 and 3.2, and Lemma 2.2(c)we get

λess0 (L) lim infk→∞

E(fk)

‖fk‖2 α2 lim

k→∞‖gk‖2‖fk‖2

= α2.

Therefore, λess0 (L) μ2/4.

3.4. A more general non-local estimate

Assume that a closed, symmetric, non-negative form (EJ ,DJ ) is given by EJ(f) =∫X×X\d(f(x) − f(y))2 dJ(x, y) for a measure J such as introduced in Subsection 3.2. Let L be

the positive self-adjoint operator associated to (EJ ,DJ ).

Theorem 3.3. Assume that ρ satisfies (♣) and fr,x,α ∈ DJ for all r 0, x ∈ X and α >μ/2. Then,

λ0(L) μ2

4and λess0 (L) μ2

4

if m(⋃Br(x0)) = ∞ for x0 used to define μ.

If the jump size is in [δ, 1] for some 0 δ 1, then

λ0(L) 2(eμ/2 − 1)2

δ2 eμ + 1and λess0 (L) 2(eμ/2 − 1)2

δ2 eμ + 1

if m(⋃Br(x0)) = ∞ for x0 used to define μ.

Proof. The proof follows along the same lines as the proof of the main theorem,Theorem 1.1, by using Proposition 2.1, Lemmas 2.2 and 3.2.

4. Applications

4.1. Weighted graphs

In this section, we derive consequences of Theorems 1.1 and 3.3 for graphs. We briefly introducethe setting and refer for more background to [18].

Let X be a countable discrete set. Every Radon measure of full support on X is givenby a function m : X → (0,∞). Then, L2(X,m) is the space 2(X,m) of m-square summablefunctions with norm ‖u‖ = (

∑x u(x)2m(x))1/2, u ∈ 2(X,m). From [18, Theorem 7], it can

be seen that all regular Dirichlet forms on (X,m) without killing term are determined by asymmetric map b : X ×X → [0,∞) with vanishing diagonal that satisfies

∑

y∈X

b(x, y) < ∞ for all x ∈ X,


and gives rise to a measure J = 1/2b on X ×X \ d (as measures on discrete spaces aredetermined by non-negative functions). The ‘12 ’ stems from the convention that we considereach edge only once in the form.

The map b can then be interpreted as a weighted graph with vertex set X. Namely, thevertices x, y ∈ X are connected by an edge with weight b(x, y) if b(x, y) > 0. In this case,we write x ∼ y. A graph is called connected if, for all x, y ∈ X, there are vertices xi ∈ X,i = 1, . . . , n such that x = x0 ∼ x1 ∼ · · · ∼ xn = y. We call (x0, x1, . . . , xn) a path connectingx and y. We say the graph is locally finite if for each x ∈ X the number of y ∈ X such thatx ∼ y is finite.

Let a map E : 2(X,m) → [0,∞] be given by

E(u) =1

2

∑

x,y∈X

b(x, y)(u(x) − u(y))2.

The regular Dirichlet form E associated to J is the restriction of E to Cc(X)‖·‖E

. Moreover, let

Emax = E |Dmax , Dmax = u ∈ 2(X,m) | E(u) < ∞,which is also a Dirichlet form that is, in general, not regular. We denote the operator arisingfrom E by L and the operator arising from Emax by Lmax.

Let ρ be an intrinsic metric on X. In the context of graphs, this is equivalent to (♣) (see[5, Lemma 4.7, Theorem 7.3]) which reads as

1

2

∑

y∈X

b(x, y)ρ(x, y)2 m(x) for all x ∈ X.

For simplicity, we restrict ourselves to the case when ρ takes values in [0,∞). (Otherwise,we can easily consider the graph componentwise.)

Remark 4.1. Very often it is convenient to consider intrinsic metrics which satisfy∑y∈X b(x, y)ρ(x, y)2 m(x) for all x ∈ X (that is, we drop the 1

2 on the left-hand side).For example, in [14, 15] an explicit example of such a metric ρ is given, for x, y ∈ X, by

ρ(x, y) := infl(x0, . . . , xn) | n 1, x0 = x, xn = y, xi ∼ xi−1, i = 1, . . . , n,where the length l is given by l(x0, . . . , xn) =

∑ni=1 minDeg(xi)

−1/2,Deg(xi−1)−1/2 andDeg(z) =

∑w b(z, w)/m(z) is a generalized vertex degree. In this case, all estimates in the

theorem below can be divided by 2.

In general, it is hard to determine whether the distance balls with respect to a certain metricare compact, meaning finite in the original topology. However, we always have a statement forthe operator Lmax related to Emax.

Theorem 4.2. Assume that b is connected and m(X) = ∞. Then,

λ0(Lmax) μ2

4and λess0 (Lmax) μ2

4.

If ρ(x, y) ∈ [δ, 1] for all x ∼ y, then

λ0(Lmax) 2(eμ/2 − 1)2

δ2eμ + 1and λess0 (Lmax) 2(eμ/2 − 1)2

δ2 eμ + 1.

Remark 4.3. If the assumption on the intrinsic metric in the theorem above is posedwithout the 1

2 on the left-hand side, then all estimates can be divided by 2.


Proof. Let Brk(x0), (respectively, Brk(xk)) be a sequence of distance balls thatrealizes μ (respectively, μ), that is, μ = lim infk→∞ r−1

k logm(Brk(x0)) (respectively, μ =lim infk→∞ r−1

k logm(Brk(xk))). If the measure of Brk(x0) (respectively, Brk(xk)) is infi-nite for some k, then μ = ∞ (respectively, μ = ∞) and we are done. Otherwise,frk,x0,α, grk,x0,α ∈ 2(X,m) (respectively, frk,xk,α, grk,xk,α ∈ 2(X,m)) and frk,x0,α ∈ Dmax

(respectively, frk,xk,α ∈ Dmax) by Lemma 3.2. Thus, the statement follows directly fromTheorem 3.3.

If we know more about either the measure or the metric structure, then we can say somethingabout the operator L. This is the case under either of the following additional assumptions:

(A) every infinite path of vertices has infinite measure;(B) ρ is an intrinsic path metric with bounded jump size on a locally finite graph such that

(X, ρ) is metrically complete.

In particular, (A) is satisfied if infx∈X m(x) > 0 and (B) is satisfied if all infinite geodesics haveinfinite length.

Corollary 4.4. Assume that either (A) or (B) is satisfied. Then, the statement ofTheorem 4.2 holds for L = Lmax.

Proof. By Keller and Lenz [18, Theorem 6], assumption (A) implies E = Emax andL = Lmax. Moreover, by Huang, Keller, Masamune and Wojciechowski [16, Theorem 2],assumption (B) also implies E = Emax and L = Lmax.

Remark 4.5. Under the slightly stronger assumption that connected infinite sets haveinfinite measure we can prove the corollary directly. Namely, if one of the relevant distanceballs is infinite, then it has infinite measure and the exponential volume growth is infinite. Inthe other case, the corollary follows from Theorem 3.3.

We also recover the result of [6] which already covers [3, 25]. In their very particular situation,m is the vertex degree and b takes values in 0, 1. The natural graph distance d is given asthe minimum length of a path of edges connecting two vertices where the length is the numberof edges contained in the path.

Corollary 4.6 (Normalized Laplacians). Let b be a connected weighted graph over (X,n),with the measure n(x) =

∑y∈X b(x, y), x ∈ X and let d be the natural graph metric. Then,

λ0(L) 1 − 2eμ/2/(1 + eμ) and λess0 (L) 1 − 2eμ/2/(1 + eμ).

Proof. Clearly, L is a bounded operator and thus L = Lmax. Moreover, the natural graphmetric is an intrinsic metric for 2L and its jump size in exactly 1. Thus, the statement followsfrom Theorem 4.2.

4.2. Standard graphs and the natural graph distance

Let b : X ×X → 0, 1 and m ≡ 1. We call such a graph standard. In this case, the operator Lbecomes the graph Laplacian Δ acting on D(Δ) = ϕ ∈ 2(X) | (x → ∑

y∼x(ϕ(x) − ϕ(y))) ∈


2(X) (see [18, 30]) as

Δϕ(x) =∑

y∼x

(ϕ(x) − ϕ(y)),

where x ∼ y means that b(x, y) = 1. By m ≡ 1, we have that m(A) = |A| for all A ⊆ X. Forsimplicity, we assume that the graph is connected and infinite.

Theorem 4.7. Let d be the natural graph distance and Bdr (x0) = x ∈ X | d(x, x0) r

for some x0 ∈ X and r 0. If

lim infr→∞

log |Bdr (x0)|

log r< 3,

then λ0(Δ) = λess0 (Δ) = 0. Moreover, if

lim supr→∞

|Bdr (x0)|r3

< ∞,

then λess0 (Δ) < ∞ and, in particular, σess(Δ) = ∅.

Remark 4.8. (a) The result above is sharp. This can be seen by the examples of antitreesdiscussed after the proof.

(b) In [9, Theorem 1.4], it is shown that less than cubic growth implies stochasticcompleteness.

(c) If the vertex degree is bounded by K, then the situation is very different: the n inCorollary 4.6 becomes deg in this case, where deg : X → N is the function assigning to avertex the number of adjacent vertices, and the corresponding normalized operator Δ acts on2(X,deg) as Δϕ(x) = (1/deg(x))

∑y∼x(ϕ(x) − ϕ(y)). Then,

λ0(Δ) λ0(Δ) Kλ0(Δ) and λess0 (Δ) λess0 (Δ) Kλess0 (Δ);

see, for example, [17]. Thus, in the case of bounded degree, the threshold again lies atsubexponential growth by Corollary 4.6 (as the measures m ≡ 1 and n = deg also give thesame exponential volume growth). Explicit estimates for the exponential volume growth ofplanar tessellations in terms of curvature can be found in [20].

(d) In the case of bounded vertex degree, we also have a threshold for the recurrence of thecorresponding random walk at quadratic volume growth; see [29, Lemma 3.12].

Let ρ be the intrinsic metric from [14] introduced above in Remark 4.1 which, in the case ofstandard graphs, is given by

ρ(x, y) = inf

n−1∑

i=0

mindeg(xi)−1/2,deg(xi+1)−1/2 | (x0, . . . , xn) is a path from x to y

.

Let Bρr = x ∈ X | ρ(x, x0) r, while Bd

r are balls with respect to the natural graph distanced for some fixed x0 ∈ X.

The proof of the theorem is based on the following lemma which is inspired by the proofof [9, Theorem 1.4]. Indeed, the second statement is taken directly from there.

Lemma 4.9. If lim infr→∞ log |Bdr |/log r = β ∈ [1, 3), then lim infr→∞ log |Bρ

r |/log r (2β/3 − β). Moreover, if lim supr→∞ |Bd

r |/r3 < ∞, then lim supr→∞(1/r) log |Bρr | < ∞.


Proof. Let Sdr = Bd

r \Bdr−1, r 0 and for convenience set Sd

−r = Bd−r = ∅ for r > 0. Let

1 α < 3 and let (rk) be an increasing sequence such that log |Bdrk

(x0)|/log rk < α for allk 0. Then,

|Bdrk

| =

rk∑

r=0

|Sdr | < rαk ,

for large k 0. For ε > 0 and k 0, set

Ak :=r ∈ [0, rk] ∩ N0||Sd

r | > α

εα(r + 1)α−1

.

We can estimate |Ak| εrk via

rαk > |Bdrk

| α

εα

∑

r∈Ak

(r + 1)α−1 α

εα

|Ak|∑

r=0

(r + 1)α−1 α

εα

∫ |Ak|

0

rα−1 dr =|Ak|αεα

.

Thus, ∣∣∣∣r ∈ [1, rk] ∩ N0

∣∣∣∣ maxi=0,1,2,3

|Sdr−i| >

α

εα(r + 1)α−1

∣∣∣∣ 4εrk

and ∣∣∣∣r ∈ [1, rk] ∩ N0

∣∣∣∣ maxi=0,1,2,3

|Sdr−i| α

εα(r + 1)α−1

∣∣∣∣ (1 − 4ε)rk.

As deg |Sdr−1 ∪ Sd

r ∪ Sdr+1| on Sd

r , we get |Dk| (1 − 4ε)rk, where

Dk :=

r ∈ [1, rk] ∩ N0

∣∣∣∣deg 3α

εα(r + 1)α−1 on Sd

r−2 ∪ Sdr−1

.

Hence, for r ∈ Dk and x ∈ Sdr−2, y ∈ Sd

r−1,

ρ(x, y) c(r + 1)−(α−1)/2 with c =√εα/3α.

Since any path from x0 to Sdrk

contains such edges, we have, for any x ∈ Sdrk

,

ρ(x0, x) c∑

r∈Dk

(r + 1)−(α−1)/2 c

rk∑

r=4εrk+2

r−(α−1)/2 c

∫ rk4εrk+2

r−(α−1)/2 dr C0r(3−α)/2k ,

with C0 > 0 for ε > 0 chosen sufficiently small and rk large. Let Rk := C0r(3−α)/2k and C :=

C−2α/(3−α)0 . Then, Bρ

Rk⊆ Bd

rkand, since |Bd

rk| =

∑rkr=0 |Sd

r | < rαk , we conclude

|BρRk

| |Bdrk

| < rαk CR2α/(3−α)k .

Thus, the first statement follows. The second statement is shown in the proof of[9, Theorem 1.4].

Proof of Theorem 4.7. In the case where the polynomial growth is strictly less than cubicwe get by the lemma above that μ = 0 with respect to the intrinsic metric ρ, and in the casewhere it is less than cubic we still have μ < ∞. Thus, the statement follows from Corollary 4.4,where (A) is clearly satisfied since m ≡ 1.

Let us discuss the example of antitrees which show the sharpness of the result. They werefirst introduced in [32] and further studied in [1, 19].

Example 4.10. An antitree is a spherically symmetric graph, where a vertex in the rthsphere is connected to all vertices in the (r + 1)th sphere for r 0, and there are no horizontal


edges. Thus, an antitree is characterized by a sequence (sr) taking values in N which encodesthe number of vertices in the sphere Sd

r = Bdr \Bd

r−1.Stronger growth than cubic. In [19, Corollary 6.6], it is shown that if an antitree has more

than cubic polynomial volume growth (that is, r3+εforε > 0), then λ0(Δ) > 0 and σess(Δ) = ∅.Indeed, with respect to the intrinsic metric ρ, these antitrees have finite diameter and thusμ = ∞; see [14, Example 4.3.2].

Cubic growth. If the distance spheres of an antitree satisfy |Sdr | = (r + 1)2, then |Bd

r | ∼(r + 1)3. Moreover, the function ϕ which takes the value r2 on vertices of the (r − 1)th sphere,r 1, is a positive generalized super-solution for Δ to the value 2, that is, Δϕ 2ϕ. Thus,by a discrete Allegretto–Piepenbrink theorem (see [31, Theorem 4.1] or [11, Theorem 3.1]) itfollows that λ0(Δ) 2. By Theorem 4.7, it follows that 2 λess0 (Δ) < ∞.

Weaker growth than cubic. In this case, Theorem 4.7 shows λ0(Δ) = λess0 (Δ) = 0.

Acknowledgements. The authors are grateful to Jozef Dodziuk and Daniel Lenz for theircontinued support and for generously sharing their knowledge.

References

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10. G. Grillo, ‘On Persson’s theorem in local Dirichlet spaces’, Z. Anal. Anwend. 17 (1998) 329–338.11. S. Haeseler and M. Keller, ‘Generalized solutions and spectrum for Dirichlet forms on graphs’, Random

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Sebastian Haeseler and Matthias KellerMathematisches InstitutFriedrich Schiller Universitat JenaD-07743 JenaGermany

sebastian·haeseler@uni-jena·demkeller@ma·huji·ac·il

Radoslaw K. WojciechowskiYork College of the City University of New

YorkJamaica, NY 11451USA

rwojciechowski@gc·cuny·edu

CHAPTER 12

F. Bauer, B. Hua, M. Keller, On the lp spectrumof Laplacians on graphs, Advances in

Mathematics 248 (2013), 717-735.

273

Available online at www.sciencedirect.com

ScienceDirect

Advances in Mathematics 248 (2013) 717–735www.elsevier.com/locate/aim

On the lp spectrum of Laplacians on graphs

Frank Bauer a,b,∗, Bobo Hua b, Matthias Keller c

a Department of Mathematics, Harvard University, One Oxford Street, Cambridge, MA 02138, USAb Max-Planck-Institut für Mathematik in den Naturwissenschaften, Inselstr. 22, 04103 Leipzig, Germany

c Einstein Institute of Mathematics, The Hebrew University of Jerusalem, Jerusalem 91904, Israel

Received 29 November 2012; accepted 2 May 2013

Communicated by Andreas Dress

Abstract

We study the p-independence of spectra of Laplace operators on graphs arising from regular Dirichletforms on discrete spaces. Here, a sufficient criterion is given solely by a uniform subexponential growth con-dition. Moreover, under a mild assumption on the measure we show a one-sided spectral inclusion withoutany further assumptions. We study applications to normalized Laplacians including symmetries of the spec-trum and a characterization for positivity of the Cheeger constant. Furthermore, we consider Laplacians onplanar tessellations for which we relate the spectral p-independence to assumptions on the curvature.© 2013 Elsevier Inc. All rights reserved.

Keywords: p-spectrum; Discrete Laplace operator; Regular Dirichlet forms; Cheeger constant; Planar tessellation

1. Introduction

In [41,42] Simon conjectured that the spectrum of a Schrödinger operator acting on Lp(RN)

is p-independent. Hempel and Voigt gave an affirmative answer in [29] for a large class of poten-tials. Later this result was generalized in various ways. Sturm [43] showed p-independence of thespectra for uniformly elliptic operators on a complete Riemannian manifold with uniform subex-ponential volume growth and a lower bound on the Ricci curvature. Moreover, Arendt [1] provedp-independence of the spectra of uniformly elliptic operators in RN with Dirichlet or Neumann

* Corresponding author at: Department of Mathematics, Harvard University, One Oxford Street, Cambridge, MA02138, USA.

E-mail addresses: [email protected] (F. Bauer), [email protected] (B. Hua), [email protected](M. Keller).

0001-8708/$ – see front matter © 2013 Elsevier Inc. All rights reserved.http://dx.doi.org/10.1016/j.aim.2013.05.029

718 F. Bauer et al. / Advances in Mathematics 248 (2013) 717–735

boundary conditions under the assumption of upper Gaussian estimates for the correspondingsemigroups. While the proof strategies of Sturm and Arendt are rather similar to the one usedby Hempel and Voigt, Davies [11] gave a simpler proof of the p-independence of the spectrumunder the stronger assumption of polynomial volume growth and Gaussian upper bounds usingthe functional calculus developed in [10]. In recent works p-independence of spectral bounds areproven in the context of conservative Markov processes [40,45] and Feynman–Kac semigroups[6,13,46]. See also [24] for Laplace operators on graphs with finite measure.

In this paper, we prove p-independence of spectra for Laplace operators on graphs underthe assumption of uniform subexponential volume growth. This question was brought up in[12, p. 378] by Davies. Our framework are regular Dirichlet forms on discrete sets as introducedin [39]. While our result is similar to the one of Sturm [43] for elliptic operators on manifolds,we do not need to assume any type of lower curvature bounds nor any type of bounded geometry.However, in various classical examples, such as Laplacians with standard weights, this disparityis resolved by the fact that uniform subexponential growth implies bounded geometry in somecases. In further contrast to [43], we do not assume any uniformity of the coefficients in the di-vergence part of the operator such as uniform ellipticity and, additionally, we allow for positivepotentials (in general potentials bounded from below).

We overcome the difficulties resulting from unbounded geometry, by the use of intrinsic met-rics. While this concept is well established for strongly local Dirichlet forms [44] it was onlyrecently introduced for general regular Dirichlet forms by Frank, Lenz and Wingert in [20].Since then, this concept already proved to be very effective for the analysis on graphs, see [3,18,19,25,28,33–35] where it also appears under the name adapted metrics. Moreover, we employrather weak heat kernel estimates (with the log term instead of a square) by Folz [18], whichis a generalization of [9] by Davies. These weak estimates turn out to be sufficient to prove thep-independence. Of course, the condition on uniform subexponential growth is always expressedwith respect to an intrinsic metric.

Another result of this paper is the inclusion of the 2-spectrum in the p-spectrum under theassumption of lower bounds on the measure only.

As applications we discuss the normalized Laplace operator, for which we prove severalbasic properties of the p spectra such as certain symmetries of the spectrum. Moreover, wediscuss consequences of p-independence on the Cheeger constant and give an example ofp-independence and superexponential volume growth. Finally, we consider the case of planartessellations which relates curvature bounds to the volume growth. In particular, we use suchcurvature conditions to recover results of Sturm [43] in the setting of planar tessellations.

The paper is organized as follows. In the next section we introduce the set up and present themain results. In Section 3 we show several auxiliary results in order to prove the main results inSections 4 and 5. Applications to normalized Laplacians are considered in Section 6. The finalsection, Section 7, is devoted to planar tessellations and consequences of curvature bounds onthe volume growth and p-independence.

2. Set up and main results

2.1. Graphs

Assume that X is a countable set equipped with the discrete topology. A strictly positivefunction m : X → (0,∞) gives a Radon measure on X of full support via m(A) = ∑

x∈A m(x)

for A ⊆ X, so that (X,m) becomes a discrete measure space.

F. Bauer et al. / Advances in Mathematics 248 (2013) 717–735 719

A graph over (X,m) is a pair (b, c). Here, c : X → [0,∞) and b : X × X → [0,∞) is asymmetric function with zero diagonal that satisfies∑

y∈X

b(x, y) < ∞, for x ∈ X.

We say x and y are neighbors or connected by an edge if b(x, y) > 0 and we write x ∼ y.For convenience we assume that there are no isolated vertices, i.e., every vertex has a neighbor.We call b locally finite if each vertex has only finitely many neighbors. The function c can beinterpreted either as one-way-edges to infinity or a potential or a killing term.

The normalizing measure n : X → (0,∞) given by

n(x) =∑y∈X

b(x, y), for x ∈ X,

often plays a distinguished role. In the case where b : X × X → 0,1, n(x) gives the number ofneighbors of a vertex x. If n/m M for some fixed M > 0, then we say the graph has boundedgeometry.

2.2. Intrinsic metrics and uniform subexponential growth

By a pseudo metric we understand a function d : X × X → [0,∞) that is symmetric, haszero diagonal and satisfies the triangle inequality. Following [20], we call a pseudo metric d anintrinsic metric for a graph b on (X,m) if∑

y∈X

b(x, y)d(x, y)2 m(x), for all x ∈ X.

For example one can always choose the path metric induced by the edge weights w(x,y) =((m/n)(x) ∧ (m/n)(y))

12 , for x ∼ y, cf. e.g. [33]. Moreover, we call

s := supd(x, y)

∣∣ x, y ∈ X, x ∼ y

the jump size of d . Note that the natural graph metric dn (i.e., the path metric with weightsw(x,y) = 1 for x ∼ y) is intrinsic if and only if m n. However, in the case of bounded geom-etry, i.e., n/m M for some fixed M > 0, the metric dn/

√M (which is equivalent to dn) is an

intrinsic metric.Throughout the paper we assume that d is an intrinsic metric with finite jump size. For

the remainder of the paper, we refer to the quintuple (X,b, c,m,d) whenever we speak of thegraph.

We denote the distance balls centered at a vertex x ∈ X with radius r 0 by Br(x) :=y ∈ X | d(x, y) r. Similar to [43], we say the graph has uniform subexponential growthif for all ε > 0 there is Cε > 0 such that

m(Br(x)

) Cεe

εrm(x), for all x ∈ X, r 0.

In Section 3.1 we discuss some implications of this assumption.

2.3. Dirichlet forms and graph Laplacians

Denote by Cc(X) the space of complex valued functions on X with compact support. Thep-spaces are given by


p := p(X,m) := f : X → C

∣∣ ‖f ‖p < ∞, p ∈ [1,∞],

where

‖f ‖∞ := supx∈X

∣∣f (x)∣∣, and ‖f ‖p :=

( ∑x∈X

∣∣f (x)∣∣pm(x)

) 1p

, p ∈ [1,∞).

Note that ∞(X,m) does not depend on m.For p ∈ [1,∞], let the Hölder conjugate be denoted by p∗, that is 1

p+ 1

p∗ = 1. We denote the

dual pairing of f ∈ p(X,m), g ∈ p∗(X,m) by

〈f,g〉 :=∑x∈X

f (x)g(x)m(x),

which becomes a scalar product for p = 2. We define the sesqui-linear form Q with domainD(Q) ⊆ 2 by

Q(f,g) = 1

2

∑x,y∈X

b(x, y)(f (x) − f (y)

)(g(x) − g(y)

) +∑x∈X

c(x)f (x)g(x),

D(Q) = Cc(X)‖·‖Q,

where ‖ · ‖Q = (Q(·) + ‖ · ‖22)

12 and Q(f ) = Q(f,f ). The form Q is a regular Dirichlet form

on 2(X,m), see [23,39] and for complexification of the forms, see [27, Appendix B]. The cor-responding positive selfadjoint operator L = L2 on 2(X,m) acts as

Lf (x) = 1

m(x)

∑y∈X

b(x, y)(f (x) − f (y)

) + c(x)

m(x)f (x).

Let L be the extension of L to

F =f : X → C

∣∣∣ ∑y∈X

b(x, y)∣∣f (y)

∣∣ < ∞ for all x ∈ X

.

We have Cc(X) ⊆ D(L) if (and only if) LCc(X) ⊆ 2(X,m), see [39, Theorem 6]. In particular,this can easily seen to be the case if the graph is locally finite or if infx∈X m(x) > 0. If m = n andc ≡ 0, then L is referred to as the normalized Laplacian.

Moreover, L = L2 gives rise to the resolvents Gα = (L − α)−1, α < 0, and the semigroupsTt = e−tL, t 0. These operators, −αGα and Tt , are positivity preserving and contractive (asQ is a Dirichlet form), and therefore extend consistently to operators on p(X,m), p ∈ [1,∞](either by monotone convergence or by density of 2 ∩ p in p , p ∈ [1,∞) and taking the dualoperator on 1 to get the operator on ∞). The semigroups are strongly continuous for p < ∞.See [8, Theorem 1.4.1] for a proof of these facts.

We denote the positive generators of Gα and Tt on p by Lp , p ∈ [1,∞), and the dual op-erator L1

∗ of L1 on ∞ by L∞ (which normally does not have dense domain in ∞). By [39,Theorem 9] we have that Lp , p ∈ [1,∞], are restrictions of L. Moreover, Lp are bounded op-erators with norm bound 2C if (n + c)/m C, see e.g. [38, Theorem 11] or [27, Theorem 9.3].Hence, bounded geometry is equivalent to boundedness of the operators for c ≡ 0. We denote thespectrum of the operator Lp by σ(Lp) and the resolvent set by ρ(Lp) = C \ σ(Lp), p ∈ [1,∞].By duality σ(Lp) = σ(Lp∗), p ∈ [1,∞].

Throughout this paper C always denotes a constant that might change from line to line.


2.4. Main results

In this section, we state the main theorems of this paper. The first result is a discrete versionof Sturm’s theorem [43], whose proof is given in Section 4.

Theorem 2.1. Assume the graph has uniform subexponential growth with respect to an intrinsicmetric with finite jump size. Then, for any p ∈ [1,∞],

σ(Lp) = σ(L2).

Remark.

(a) Other than [43], we do not assume any type of bounded geometry or any types of lowerbounds on the curvature. For a discussion and examples see Section 3.1.

(b) Sometimes loops in the graph are modeled by non-vanishing diagonal of b. However, theassumption that b has zero diagonal has no influence on our main results above as possiblenon-vanishing diagonal terms do not enter the operators. Such loops only have an effecton n. Thus, when b had non-vanishing diagonal, then one only had to be careful wheneverone chooses m = n.

Clearly, we can also allow for potentials c such that c/m is only bounded from below (asadding a positive constant shifts the p spectra of the operators simultaneously).

The following theorem shows that under an assumption on the measure one spectral inclusionholds without any volume growth assumptions.

Theorem 2.2. If m is such that infx∈X m(x) > 0, then, for any p ∈ [1,∞],σ(L2) ⊆ σ(Lp).

The proof of Theorem 2.2 is given in Section 5.

3. Preliminaries

In this section we collect some results and facts that will be used for the proof of Theorem 2.1.Moreover, in the first subsection we discuss the relation of uniform subexponential growth andbounded geometry.

3.1. Consequences of uniform subexponential growth

Lemma 3.1. Assume the graph has uniform subexponential growth. Then, for all ε > 0 there isC > 0 such that:

(a) m(x) Ceεd(x,y)m(y), for all x, y ∈ X.(b) #Br(x) Ceεr , for all r 0, where #Br(x) denotes the number of vertices in Br(x).(c)

∑y∈X e−εd(x,y) C, for all x ∈ X.


Proof. To prove (a) let x, y ∈ X. Using the uniform subexponential growth assumption andx ∈ Bd(x,y)(y) yields m(x) m(Bd(x,y)(y)) Ceεd(x,y)m(y). Turning to (b), let x ∈ X andr 0. We obtain using (a) and the uniform subexponential growth assumption

#Br(x) =∑

y∈Br (x)

m(y)/m(y) Ceεr m(Br(x))

m(x) C2e2εr .

The final statement (c) follows also by direct calculation using (b) (with ε1)

∑y∈X

e−εd(x,y) =∞∑

r=1

∑y∈Br(x)\Br−1(x)

e−εd(x,y) ∞∑

r=1

e−ε(r−1)#Br(x) C

∞∑r=1

e(ε1−ε)r .

Hence, choosing ε1 = ε/2 yields the statement. Remark.

(a) Lemma 3.1 (b) implies finiteness of distance balls. On the other hand, finite jump size s

implies that for each vertex x all neighbors of x are contained in Bs(x). Hence, graphs withuniform subexponential growth and finite jump size are locally finite.

(b) Finiteness of distance balls has strong consequences on the uniqueness of selfadjoint exten-sions. In particular, by [35, Corollary 1] implies that Q is the maximal form on 2 and thatthe restriction of L2 to Cc(X) (whenever Cc(X) ⊆ D(L2)) is essentially selfadjoint.

In the following we discuss examples to clarify the relation between uniform subexponentialgrowth and bounded geometry in the discrete setting.

Recall that we speak of bounded geometry if n/m is a bounded function which is a naturaladaption to the situation of weighted graphs. In Example 3.2 below, we show that there are graphswith uniform subexponential growth and unbounded geometry. For completeness we also givean example of bounded geometry and exponential growth which is certainly well known.

Example 3.2.

(a) Uniform subexponential growth and unbounded geometry. Let X = N, m ≡ 1, c ≡ 0and consider b such that b(x, y) = 0 for |x − y| = 1, b(x, x + 1) = x for x ∈ 4N andb(x, x +1) = 1 otherwise. Clearly, (n/m)(x) = n(x) = x +1 for x ∈ 4N and, thus, the graphhas unbounded geometry. In particular, Lp is unbounded for all p ∈ [1,∞]. Moreover, let

d be the path metric induced by the edge weights w(x,x + 1) = (n(x) ∨ n(x + 1))− 12 . We

obtain that d(x, y) (|x − y| − 3)/(4√

2 ) for all x, y ∈ X. Hence, m(Br(x)) = #Br(x) √2(8r + 6) which implies uniform subexponential growth.

(b) Exponential growth and bounded geometry. Take a regular tree, c ≡ 0, set b to be one on theedges and zero otherwise and let m ≡ 1. This graph has bounded geometry but is clearly ofexponential growth.

It is apparent that the graph in Example 3.2 (a) above has bounded combinatorial vertex degreeand the unbounded geometry is induced by the edge weights. So, one might wonder whetherthere are also examples with unbounded combinatorial vertex degree which is the criterion forunbounded geometry in the classical setting. The proposition below shows that this is impossible


under the assumptions of uniform subexponential growth and finite jump size. Recall that by theremark below Lemma 3.1 we already know that the graph must be locally finite.

The combinatorial vertex degree deg is the function that assigns to each vertex the number ofneighbors, that is deg(x) = #y ∈ X | b(x, y) > 0, x ∈ X.

Proposition 3.3. If the graph has uniform subexponential growth with respect to a metric withfinite jump size s, then the combinatorial vertex degree is bounded.

Proof. Suppose the graph has unbounded vertex degree, i.e., there is a sequence of vertices (xn)

such that deg(xn) n2 for all n 1. We show that there is a sequence of vertices zn suchthat m(Bs(zn))/m(zn) is unbounded and thus the graph does not have uniform subexponentialgrowth.

If, for n 1, there is a neighbor yn of xn such that m(yn) m(xn)/√

deg(xn), then we esti-mate using xn ∈ Bs(yn)

m(Bs(yn))

m(yn) m(xn)

m(yn)

√deg(xn) n.

We set zn = yn in this case. If, on the other hand, m(y) m(xn)/√

deg(xn) for all neighbors y

of xn, then

m(Bs(xn))

m(xn) 1

m(xn)deg(xn)

m(xn)√deg(xn)

√

deg(xn) n

and set zn = xn in this case. Hence, we have proven the claim. Corollary 1. Assume there is D > 0 such that b D and m 1/D. Then, uniform subexponen-tial growth with respect to a metric with finite jump size implies bounded geometry.

Proof. One simply observes that n/m D2 deg and the statement follows from the propositionabove.

The lemma and the corollary above mean for the standard Laplacians pϕ(x) =∑y∼x(ϕ(x) − ϕ(y)) on p(X,1) and

(n)p ϕ(x) = 1

deg(x)

∑y∼x(ϕ(x) − ϕ(y)) on p(X,deg),

that uniform subexponential growth implies bounded geometry, both in the sense of boundedn/m and also in the sense of bounded deg.

3.2. Lipschitz continuous functions

We denote by Lip∞ε the real valued bounded Lipschitz continuous functions with Lipschitz

constant ε > 0, i.e.,

Lip∞ε :=

ψ : X → R∣∣ ψ(x) − ψ(y) εd(x, y), x, y ∈ X

∩ ∞(X,m).

Lemma 3.4. Let ε > 0 and let s be the jump size of d . Then, for all ψ ∈ Lip∞ε :

(a) eψ is a bounded Lipschitz continuous function, in particular, eψD(Q) = D(Q).(b) |1 − eψ(x)−ψ(y)| εeεsd(x, y), for x ∼ y.(c) |(e−ψ(x) − e−ψ(y))(eψ(x) − eψ(y))| 2ε2eεsd(x, y)2, for x ∼ y.


Proof. The first statement of (a) follows from mean value theorem, that is for any x, y ∈ X wehave ∣∣eψ(x) − eψ(y)

∣∣ ∣∣ψ(x) − ψ(y)

∣∣e‖ψ‖∞ εd(x, y)e‖ψ‖∞ .

Now, eψD(Q) ⊆ D(Q) is a consequence of [25, Lemma 3.5]. The other inclusion follows sincee−ψ is also bounded and Lipschitz continuous. Similarly, we get (b) using the Taylor expansionof the exponential function

∣∣1 − eψ(x)−ψ(y)∣∣ =

∑k1

(ψ(x) − ψ(y))k

k! εd(x, y)∑k1

(εs)k−1

k! εd(x, y)eεs,

and, similarly, using |(e−ψ(x) − e−ψ(y))(eψ(x) − eψ(y))| = 2∑

k∈2N(ψ(x)−ψ(y))k

k! we get (c). 3.3. Kernels

Let A : D(A) ⊆ p → q , p,q ∈ [1,∞] be a densely defined linear operator. We denote by‖A‖p,q the operator norm of A, i.e.

‖A‖p,q = supf ∈D(A), ‖f ‖p=1

‖Af ‖q .

Note that any such operator A : D(A) ⊆ p → q , p < ∞, with Cc(X) ⊆ D(A) admits a kernelkA : X × X → C such that

Af (x) =∑x∈X

kA(x, y)f (y)m(y)

for all f ∈ D(A), x ∈ X, which can be obtained by

kA(x, y) = 1

m(x)m(y)〈A1y,1x〉,

where 1v(w) = 1 if w = v and 1v(w) = 0 otherwise.We recall the following well-known lemma which shows that the operator norm of A : p →

q can be estimated by its integral kernel.

Lemma 3.5. Let p ∈ [1,∞) and let A be a densely defined linear operator with Cc(X) ⊆D(A) ⊆ p . Then:

(a) ‖A‖p,q (∑

y ‖kA(·, y)‖p∗q m(y))

1p∗ for q < ∞ and p ∈ (1,∞), and ‖A‖1,q

supy ‖kA(·, y)‖q for q < ∞.(b) ‖A‖p,∞ supx ‖kA(x, ·)‖p∗ and equality holds if p = 1.

Proof. Statement (a) follows from the fact ‖f ‖p = supg∈p∗, ‖g‖p∗=1〈g,f 〉 and twofold appli-

cation of Hölder inequality. The first part of (b) follows simply from Hölder inequality. For thesecond part note that ‖A‖1,∞ supx,y∈X |A1y(x)|/m(y).


3.4. Heat kernel estimates

We denote the kernel of the semigroup Tt , t 0 by pt . As the semigroups are consistenton p , p ∈ [1,∞], i.e., they agree on their common domains, the kernel pt does not dependon p.

The following heat kernel estimate will be the key to p-independence of spectra of Lp . It isproven in [18], based on [9], for locally finite graphs and c ≡ 0. However, on the one hand, localfiniteness is not used in [18] for this result and, on the other hand, the remark below Lemma 3.1shows that we are in the local finite situation anyway whenever we assume uniform subexponen-tial growth. We conclude the statement for c 0 by a Feynman–Kac formula.

Lemma 3.6. We have for all t 0 and x, y ∈ X

pt(x, y) (m(x)m(y)

)− 12 e−d(x,y) log d(x,y)

2et .

Proof. Denote the semigroup of the graph (b,0) by T(0)t and the kernel by p

(0)t and correspond-

ingly for (b, c) by Tt and pt .For c ≡ 0 the estimate above is found in [18, Theorem 2.1] for p

(0)t . Now, by a Feynman–Kac

formula, see e.g. [14,26], we have

pt (x, y) = Ttδy(x) = Ex

[e− ∫ t

0cm

(Xs ) dsδy(Xt )] Ex

[δy(Xt )

] = T(0)t δy(x) = p

(0)t (x, y),

where δy = 1y/m(y). This proves the claim. By basic calculus, we obtain the following heat kernel estimate.

Lemma 3.7. For all β > 0 there exists a constant C(β) such that for all t 0, x, y ∈ X

pt(x, y) (m(x)m(y)

)− 12 e−βd(x,y)+C(β)t .

Proof. Let β > 0 and r > 0 let f (r) = −r log(r/2e) + βr . Direct calculation shows that thefunction f assumes its maximum on the domain (0,∞) at the point r0 = 2eβ . In particular,setting C(β) = 2eβ yields

−d(x, y)

tlog

d(x, y)

2et −β

d(x, y)

t+ C(β),

for all t > 0 and x, y ∈ X. The statement follows now from the lemma above. 4. Proof for uniform subexponential growth

In this section we prove Theorem 2.1 following the strategy of [43]. The proof is divided intoseveral lemmas and as always we assume that d is an intrinsic metric with finite jump size s.

Lemma 4.1. For every compact set K ⊆ ρ(L2) there is ε > 0 and C < ∞ such that for all z ∈ K

and all ψ ∈ Lip∞ε∥∥e−ψ(L2 − z)−1eψ

∥∥2,2 C.


Proof. Let ε > 0 and ψ ∈ Lip∞ε . By Lemma 3.4 we have e−ψD(Q) = eψD(Q) = D(Q). Let

Qψ be the (not necessarily symmetric) form with domain D(Qψ) = D(Q) acting as

Qψ(f,g) := Q(e−ψf, eψg

) − Q(f,g).

Application of Leibniz rule yields, for f ∈ D(Qψ),

∣∣Qψ(f,f )∣∣ = 1

2

∑x,y∈X

b(x, y)∣∣f (y)

∣∣2(e−ψ(x) − e−ψ(y)

)(eψ(x) − eψ(y)

)

+ 1

2

∑x,y∈X

b(x, y)f (y)(f (x) − f (y)

)(1 − eψ(y)−ψ(x)

)

+ 1

2

∑x,y∈X

b(x, y)f (y)(1 − eψ(x)−ψ(y)

)(f (x) − f (y)

).

Applying Cauchy–Schwarz inequality, Lemma 3.4 (b) and (c) and the intrinsic metric property,gives

· · · ε2C∑

x,y∈X

∣∣f (x)∣∣2

b(x, y)d(x, y)2 + 2Cε

( ∑x,y∈X

∣∣f (x)∣∣2

b(x, y)d(x, y)2) 1

2

Q(f )12

Cε2‖f ‖22 + 2Cε‖f ‖2Q(f )

12 .

Hence, the basic inequality 2ab (1/δ)a2 + δb2 for δ > 0 and a, b 0 (applied with a =Cε‖f ‖2

2 and b = Q(f )12 ) yields

∣∣Qψ(f,f )∣∣ Cε2

(1 + 1

δ

)‖f ‖2

2 + δQ(f ).

This shows that Qψ is Q bounded with bound 0. According to [36, Theorem VI.3.9] this impliesthat the form Qψ + Q is closed and sectorial. It can be checked directly that the correspondingoperator is eψL2e

−ψ with domain Dψ = eψD(L2). Moreover, for K ⊆ ρ(L2) compact, wecan choose ε, δ > 0 that 2‖(C(1 + 1/δ)ε2e2s + δL2) · (L2 − z)−1‖2,2 < 1 for all z ∈ K sinceC is a universal constant. Therefore, again by [36, Theorem VI.3.9] this implies existence ofC = C(K,ε) such that∥∥e−ψ(L2 − z)−1eψ

∥∥2,2 = ∥∥(

eψL2e−ψ − z

)−1∥∥2,2 C

for all z ∈ K and ψ ∈ Lip∞ε .

Let us recall some well-known facts about consistency of semigroups and resolvents. By [8,Theorem 1.4.1] the semigroups Tt are consistent on p . By the spectral theorem the Laplacetransform for the resolvent Gz = (L2 − z)−1

Gzf =∞∫

0

eztTtf dt,

holds for f in 2 and z ∈ w ∈ C | w < 0 (the open left half plane). By density and dualityarguments, this formula extends to f in p , p ∈ [1,∞), in the strong sense and to p = ∞ in


the weak sense. This shows that the resolvents (Lp − z)−1 are consistent on p for z ∈ w ∈ C |w < 0.

We denote by gα the kernel of the resolvent Gα = (Lp − α)−1 which is independent of p ∈[1,∞] for α < 0.

Lemma 4.2. Assume the graph has uniform subexponential volume growth. For any ε > 0 thereexists α < 0 and C < ∞ such that

(a) |gα(x, y)| C(m(x)m(y))− 12 e−εd(x,y) for all x, y ∈ X,

(b) ‖eψGαe−ψm12 ‖1,2 C for all ψ ∈ Lip∞

ε ,

(c) ‖m 12 eψGαe−ψ‖2,∞ C for all ψ ∈ Lip∞

ε .

Proof. (a) By the Laplace transform of the resolvent and Lemma 3.7, we get

gα(x, y) =∞∫

0

eαtpt (x, y) dt (m(x)m(y)

)− 12 e−εd(x,y)

∞∫0

e(α+C)t dt,

which yields the statement for α < −C.(b) By Lemma 3.5 (a), ψ ∈ Lip∞

ε , part (a) above (with ε1 2ε) and Lemma 3.1 (c)∥∥eψGαe−ψm12∥∥2

1,2 supy∈X

∥∥gα(·, y)eψ(·)−ψ(y)m(y)12∥∥2

2

C supy∈X

∑x∈X

e(2ε−2ε1)d(x,y) < ∞.

The proof of (c) works similarly using Lemma 3.5 (b). Lemma 4.3. Assume the graph has uniform subexponential volume growth. Then, (L2 − z)−2

extends to a bounded operator on p for all z ∈ ρ(L2) and p ∈ [1,∞]. Moreover, for all compactK ⊆ ρ(L2) there is C < ∞ such that for all z ∈ K and p ∈ [1,∞]∥∥(L2 − z)−2

∥∥p,p

C.

Proof. For z ∈ ρ(L2) denote by g(2)z the kernel of the squared resolvent (Gz)

2 = (L2 − z)−2.Applying the resolvent identity twice yields

(Gz)2 = (

Gα + (z − α)GαGz

)(Gα + (z − α)GzGα

) = Gα

(I + (z − α)Gz

)2Gα,

for all α < 0. Therefore,

m12 eψ(Gz)

2e−ψm12 = (

m12 eψGαe−ψ

)(I + (z − α)eψ/2Gze

−ψ/2)2(eψGαe−ψm

12),

for all ε > 0 and ψ ∈ Lip∞ε . Taking the norm ‖ · ‖1,∞ and factorizing ‖ . . .‖1,∞ ‖(. . .)‖2,∞ ×

‖(. . .)‖2,2‖(. . .)‖1,2 yields that U := m12 eψG2

ze−ψm

12 is a bounded operator 1 → ∞ by

Lemma 4.1 and Lemma 4.2 with appropriate choice of α < 0, ε > 0 and all ψ ∈ Lip∞ε . Hence,

the operator U admits a kernel kU (x, y) = (m(x)m(y))12 eψ(x)−ψ(y)g

(2)z (x, y), x, y ∈ X, and we

conclude from Lemma 3.5 (b) that∣∣g(2)z (x, y)

∣∣ C(m(x)m(y)

)− 12 eψ(y)−ψ(x).


For chosen ε > 0 and any fixed x, y ∈ X let ψ : u → ε(d(u, y) ∧ d(x, y)) and we obtain fromLemma 3.1 (a) (with ε)

∣∣g(2)z (x, y)

∣∣ C(m(x)m(y)

)− 12 e−εd(x,y) Cm(x)−1e− ε

2 d(x,y).

Thus, using Lemma 3.5 (a) and Lemma 3.1 (c), we obtain∥∥G2z

∥∥1,1 sup

y∈X

∑x∈X

∣∣g(2)z (x, y)

∣∣m(x) C supy∈X

∑x∈X

e− ε2 d(x,y) < ∞.

As G2z is bounded for p = 1 and p = 2, it follows from the Riesz–Thorin interpolation theorem

that it is bounded for p ∈ [1,2] and by duality for p ∈ [1,∞]. Lemma 4.4. If σ(Lp) ⊆ [0,∞) for all p ∈ [1,∞], then σ(L2) ⊆ σ(Lp).

Proof. The operators (Lp − z)−1 and (Lp∗ − z)−1 are consistent for z ∈ w ∈ C | w < 0 ⊆ρ(Lp) = ρ(Lp∗) by the discussion above Lemma 4.2 for all p ∈ [1,∞]. By the assumptionσ(Lp) ⊆ [0,∞) the resolvent sets are connected which yields by [30, Corollary 1.4] that(Lp − z)−1 and (Lq − z)−1 are consistent for z ∈ ρ(Lp) ∩ ρ(Lq) for p,q ∈ [1,∞]. More-over, by the standard theory [12, Lemma 8.1.3] (Lp − z)−1 and (Lp∗ − z)−1 are analytic onρ(Lp) = ρ(Lp∗). By the Riesz–Thorin theorem these resolvents can be consistently extended toanalytic 2-bounded operators, see [12, Lemma 1.4.8]. That is, as an 2-bounded operator-valuedfunction (Lp −z)−1 is analytic on ρ(Lp) which is consistent with (L2 −z)−1 on ρ(Lp)∩ρ(L2).Note that, (L2 −z)−1 is analytic on ρ(L2) which is also the maximal domain of analyticity. Thus,the statement follows by unique continuation. Proof of Theorem 2.1. We start by showing σ(Lp) ⊆ σ(L2). For the kernel g

(2)z of (L2 − z)−2

and fixed x, y ∈ X the function ρ(L2) → C, z → g(2)z (x, y) is analytic. By Lemma 4.3 we know

that for any compact K ⊆ ρ(L2) the operators (L2 −z)−2, z ∈ K are bounded on p , p ∈ [1,∞].Therefore (L2 −z)−2 is analytic as a family of p-bounded operators for z ∈ ρ(L2). On the otherhand, (Lp − z)−2 is analytic as a family of p-bounded operators with domain of analyticityρ(Lp) by [29, Lemma 3.2]. Since (L2 − z)−2 and (Lp − z)−2 agree on w ∈ C | w < 0, byunique continuation they agree as analytic p operator-valued functions on ρ(L2). As the domainof analyticity of (Lp − z)−2 is ρ(Lp), this implies ρ(L2) ⊆ ρ(Lp).

On the other hand, since σ(Lp) ⊆ σ(L2) ⊆ [0,∞) the statement σ(L2) ⊆ σ(Lp) followsfrom Lemma 4.4. 5. Proof for uniformly positive measures

In this section we consider measures that are uniformly bounded from below by a positiveconstant and prove Theorem 2.2. We notice that infx∈X m(x) > 0 implies

p ⊆ q, 1 p q ∞.

Moreover, by [39, Theorem 5] we know the domains of the generators Lp in this case explicitly,namely

D(Lp) := f ∈ p

∣∣ Lf ∈ p,


where L was defined in Section 2.3. In particular, this gives

D(Lp) ⊆ D(Lq), 1 p q ∞.

Furthermore, it can be checked directly that Cc(X) ⊆ D(Lp).

Lemma 5.1. Assume infx∈X m(x) > 0. Then, for all 1 p q ∞ and all z ∈ ρ(Lp) ∩ ρ(Lq)

the resolvents (Lp − z)−1 and (Lq − z)−1 are consistent on p = p ∩ q .

Proof. Let 1 p < q ∞ and z ∈ ρ(Lp) ∩ ρ(Lq). As D(Lp) ⊆ D(Lq) and Lp = Lq onD(Lp), we have for all f ∈ p ⊆ q

(Lq − z)(Lp − z)−1f = (Lp − z)(Lp − z)−1f = f.

Hence, (Lp − z)−1 and (Lq − z)−1 are consistent on p = p ∩ q . Proof of Theorem 2.2. Let p ∈ [1,2]. By the lemma above the resolvents (Lp − z)−1 and(Lp∗ − z)−1 are consistent for z ∈ ρ(Lp) = ρ(Lp∗) on p . By the Riesz–Thorin interpolationtheorem (Lp∗ − z)−1 is bounded on 2. We will show that (Lp∗ − z)−1 is an inverse of (L2 − z)

for z ∈ ρ(Lp) = ρ(Lp∗). So, let z ∈ ρ(Lp∗). As D(L2) ⊆ D(Lp∗), 2 ⊆ p∗and L2, Lp∗ are

restrictions of L we have for f ∈ D(L2)

(Lp∗ − z)−1(L2 − z)f = (Lp∗ − z)−1(Lp∗ − z)f = f.

Secondly, let f ∈ 2 and (fn) be such that fn ∈ p and fn → f in 2. As (Lp∗ − z)−1

is 2-bounded, (Lp∗ − z)−1fn → (Lp∗ − z)−1f , n → ∞, in 2. By the lemma above(Lp∗ − z)−1fn = (Lp − z)−1fn ∈ D(Lp) ⊆ D(L2), and, therefore,

(L2 − z)(Lp∗ − z)−1fn = (Lp∗ − z)(Lp∗ − z)−1fn = fn → f, n → ∞,

in 2. Since, L2 is closed we infer (Lp∗ − z)−1f ∈ D(L2) and (L2 − z)(Lp∗ − z)−1f = f .Hence, (Lp∗ − z)−1 is an inverse of (L2 − z) and, thus, z ∈ ρ(L2). Remark. The abstract reason behind Theorem 2.2 is that the semigroup e−tL is ultracontractive,i.e. a bounded operator from 2 to ∞, which is a consequence of e−tL being a contraction on ∞(as Q is a Dirichlet form) and the uniform lower bound on the measure. Knowing this one candeduce by duality and interpolation that e−tL is a bounded operator from p to q , p q (cf. [42,proof of Theorem B.1.1]) and employ the proof of [29, Proposition 2.1] or [30, Proposition 3.1].

6. Spectral properties of normalized Laplacians

In this section, we consider normalized Laplace operators that is we assume m = n, wheren(x) = ∑

y∈X b(x, y), x ∈ X, and c ≡ 0.

6.1. Symmetries of the spectrum

Recall that a graph is called bipartite if the vertex set X can be divided into two disjoint subsetsX1 and X2 such that every edge connects a vertex in X1 to a vertex in X2, i.e., b(x, y) > 0 for apair (x, y) ∈ X × X implies (x, y) ∈ (X1 × X2) ∪ (X2 × X1).


Theorem 6.1. Assume m = n, c ≡ 0 and let p ∈ [1,∞]. Then, σ(Lp) is included in z ∈ C ||z − 1| 1 and is symmetric with respect to R = z ∈ C | z = 0, i.e., λ ∈ σ(Lp) if and only ifλ ∈ σ(Lp). Moreover, if the graph is bipartite, then σ(Lp) is symmetric with respect to the linez ∈ C: z = 1, i.e., λ ∈ σ(Lp) if and only if (2 − λ) ∈ σ(Lp).

The proof of the second part of the theorem is based on the following lemma [12,Lemma 1.2.13].

Lemma 6.2. Let A : B → B be a bounded operator on a Banach space B. Then λ ∈ σ(A) if andonly if at least one of the following occurs:

(i) λ is an eigenvalue of A.(ii) λ is an eigenvalue of A∗, where A∗ is the dual operator of A.

(iii) There exists a sequence (fn) in B with ‖fn‖ = 1 and limn→∞ ‖Afn − λfn‖ = 0.

Proof of Theorem 6.1. Let p ∈ [1,∞]. As n/m = 1 the operator Lp is bounded on p by [38,Theorem 11] (or [27, Theorem 9.3]) with bound 2. Moreover, the operator Lp can be representedas Lp = I − Pp where Pp is the transition matrix acting as Ppf (x) = 1

n(x)

∑y b(x, y)f (y).

Direct calculation shows that ‖Pp‖p,p 1. As the spectral radius is smaller than the norm,σ(Pp) ⊆ z ∈ C | |z| 1. Now, I is a spectral shift by 1, so the first statement follows.

The first symmetry statement follows from the fact that the integral kernel of Lp is real valued.Let now X1 and X2 be a bipartite partition of X. For a function f : X → C, we denote

f = 1X1f − 1X2f,

where 1W is the characteristic function of W ⊆ X. We separate the proof into three cases accord-ing to the preceding lemma.

Case 1: λ is an eigenvalue of Lp with eigenfunction f . By direct calculation it can be checkedthat f is an eigenfunction of Lp for the eigenvalue 2 − λ.

Case 2: λ is an eigenvalue of L∗p . By the same argument as in Case 1 we get that 2 − λ is an

eigenvalue of Lp∗ . Therefore, by σ(L∗p) = σ(Lp), we conclude the result.

Case 3: λ is such that there is fn with ‖fn‖ = 1, n 1 and lim‖(Lp − λ)fn‖p = 0. It iseasy to see that ‖fn‖p = ‖fn‖p and by direct calculation it follows ‖(Lp − (2 − λ))fn‖p =‖(Lp − λ)fn‖p . Thus, the statement (2 − λ) ∈ σ(Lp) follows by Lemma 6.2 (iii). 6.2. Cheeger constants

Define the Cheeger constant α 0 to be the maximal β 0 such that for all finite W ⊆ X

βn(W) |∂W |,where |∂W | := ∑

(x,y)∈W×(X\W) b(x, y). For recent developments concerning Cheeger constantssee also [2,3].

Proposition 6.3. Assume m = n and c ≡ 0. Then, infσ(L1) = infσ(L2) if and only if α = 0.


Proof. As L∞ is bounded it is easy to see that the constant functions are eigenfunctions to theeigenvalue 0. Hence, infσ(L1) = infσ(L∞) = 0. Now, 0 = infσ(L2) is equivalent to α = 0 bya Cheeger inequality [38] (cf. [17] and [16]). Remark. The proposition can also be obtained as a consequence of [45, Theorem 3.1]. The con-siderations therein show that one direction of the result is still valid in certain situations involvingunbounded operators using the Cheeger constant defined in [3]. However, in this case it is usuallyhard to determine whether the constant functions are in the domain of L∞. Nevertheless, this isthe case if the graph is stochastically complete (see [39] for characterizations of this case).

6.3. Spectral independence and superexponential growth

This previous proposition shows that α = 0 is a necessary condition for the p-independence ofthe spectrum for p ∈ [1,∞]. However, the next proposition shows that if we exclude p = 1,∞,then we can have p-independence of the spectrum even if α > 0 or if the subexponential volumegrowth condition is not satisfied.

Define the Cheeger constant at infinity α∞ 0 to be the maximum of all β 0 such that forsome finite K ⊆ X and all finite W ⊆ X \ K and βn(W) |∂W |.

One can easily check that under the assumption of m(X) = ∞, α > 0 if and only if α∞ > 0:The inequality α α∞ is obvious. If α = 0 but α∞ > 0, the Cheeger estimates, see e.g.[22,37,38] imply 0 ∈ σ(L2), but 0 /∈ σess(L2). This, however, means that 0 is an eigenvalue whichis impossible as the constant functions are not in 2 due to m(X) = ∞, cf. [24, Theorem 6.1].Hence, the next theorem says that we have p-independence for p ∈ (1,∞) even though α > 0.

Theorem 6.4. Assume m = n and α∞ = 1. Then, σ(Lp) = σ(L2) for all p ∈ (1,∞) andthe graph has superexponential volume growth with respect to the natural graph metric, i.e.,limr→∞ 1

rlogn(Br(x)) = ∞, for all x ∈ X.

Proof. It was shown in [38] (cf. [37,22] for the unweighted case) that α∞ = 1 impliesσess(L2) = 1 which is equivalent to I − L2 being a compact operator. Since I − Lp is aconsistent family of bounded operators for p ∈ [1,∞] it follows from a well-known result byKrasnosel’skii and Persson (see [12, Theorem 4.2.14]) that I −Lp is compact for all p ∈ [2,∞).Using a theorem of Schauder (see for instance [12, Theorem 4.2.13]), we conclude that I − Lp

is compact for all p ∈ (1,∞). Now, it follows from [12, Theorem 4.2.15] that the spectrum ofLp is p-independent for all p ∈ (1,∞). The second part of the proposition follows directly from[21, Theorem 1] (for the weighted case combine [38, Theorem 19] and [28, Theorem 4.1]). Remark 6.5.

(a) For examples satisfying the assumptions of the theorem above see [37,22].(b) Under the assumption m = n and α > 0 one can show that I − L1 and I − L∞ are not

compact operators. Assume the contrary, i.e., I − L1 and I − L∞ are compact, then [12,Theorem 4.2.15] implies that the spectrum is p-independent for all p ∈ [1,∞]. This, how-ever, contradicts Proposition 6.3.


7. Tessellations and curvature

For this final section, we restrict our attention to planar tessellations and relate curvaturebounds to volume growth and p-independence. We show analogue statements to Propositions 1and 2 of [43] and discuss the case of uniformly unbounded negative curvature.

We consider graphs such that b takes values in 0,1 and c ≡ 0. The two prominent choicesfor m are either m = n or m ≡ 1. For m = n the natural graph metric dn is an intrinsic metric andfor m ≡ 1 the path metric d1 given by

d1(x, y) = infx=x0∼···∼xn=y

n∑i=1

(n(xi−1) ∨ n(xi)

)− 12 , x, y ∈ X,

is an intrinsic metric. Both metrics have the jump size at most 1, in particular, both metrics havefinite jump size. We denote the Laplacian with respect to m = n by

(n)p and with respect to

m ≡ 1 by p , p ∈ [1,∞].Note that by Theorem 2.2 we have for all p ∈ [1,∞]

σ(

(n)2

) ⊆ σ((n)

p

)and σ(2) ⊆ σ(p).

Let G be a planar tessellation, see [4,5] for background. Denote by F the set of faces anddenote the degree of a face f ∈ F , that is the number of vertices contained in face, by deg(f ).

We define the vertex curvature κ : X → R by

κ(x) = 1 − n(x)

2+

∑f ∈F, x∈f

1

deg(f ).

The following theorem is a discrete analogue of [43, Proposition 1].

Theorem 7.1. If κ 0, then it has quadratic volume growth both with respect to dn and m = n

and with d1 and m ≡ 1. In particular, σ((n)p ) = σ(

(n)2 ) and σ(p) = σ(2) for all p ∈ [1,∞].

Proof. By [32, Theorem 1.1] the volume growth for κ 0 is bounded quadratically with respectto the measure m = n and the metric dn. Moreover, it is direct to check that κ 0 implies thatthe vertex degree is bounded by 6. Hence, we have bounded geometry and thus dn and d1 areequivalent and for m ≡ 1 we have m n/6. Thus, the volume growth of the graph is quadraticallybounded also with respect to the measure m ≡ 1 and the metric d1. The ‘in particular’ is now aconsequence of Theorem 2.1. Remark. The result can be easily extended to planar tessellation with finite total curvature,i.e.,

∑x∈X |κ(x)| < ∞ or equivalently vanishing curvature outside of a finite set, see [7] and

also [15]. This can be seen as [32, Theorem 1.1] easily extends to the finite total curvature case.

The next theorem is a discrete analogue of [43, Proposition 2]. We say that a graph with mea-sure m has at least exponential volume growth with respect to a metric d if for the correspondingdistance balls Br(x) about some vertex x,

μ = lim infr→∞ inf

x∈X

1

rlogm

(Br(x)

)> 0.


Theorem 7.2. If κ < 0, then the tessellation has at least exponential volume growth, both withrespect to dn and m = n and with d1 and m ≡ 1. Moreover, infσ(

(n)1 ) = infσ(

(n)2 ) and if we

have additionally bounded geometry, then infσ(1) = infσ(2).

Proof. [31, Theorem C, Proposition 2.1] implies that if κ(x) < 0 for all x ∈ X, thensupx∈X κ(x) −1/1806 < 0. Hence, by [31, Theorem B] we have positive Cheeger constant,

α > 0. Thus, by a Cheeger inequality [17] infσ((n)2 ) α2/2 > 0 and by elementary compu-

tations (cf. [37]) we infer infσ(2) infσ((n)2 ) · infx deg(x) > 0. By [28, Corollary 4.2] this

implies at least exponential volume growth with respect to both metrics. The second statementfollows along the lines of the proof of Proposition 6.3.

We end this section by an analogue theorem of Theorem 6.4.

Theorem 7.3. If infK⊆X, finite supx∈X\K κ(x) = −∞, then σ((n)p ) = σ(

(n)2 ) and σ(p) =

σ(2) for all p ∈ (1,∞). In particular, the spectrum is purely discrete and the eigenfunctions of2 are contained in p for all p ∈ (1,∞).

Proof. By [37, Theorem 3] the curvature assumption is equivalent to pure discrete spectrumof 2. Moreover, this is equivalent to compact resolvent and compact semigroup on 2. Thus, thestatement for p follows from [8, Theorem 1.6.3]. On the other hand, the curvature assumption

implies α∞ = 1 [37], and the statement about (n)p follows from Theorem 6.4.

Acknowledgments

The first and second authors thank Jürgen Jost for many inspiring discussions on this topic.The first author acknowledges partial financial support of the Alexander von Humboldt Foun-dation and partial financial support of the NSF grant DMS-0804454 Differential Equations inGeometry. The research leading to these results has received funding from the European ResearchCouncil under the European Union’s Seventh Framework Programme (FP7/2007-2013)/ERCgrant agreement No. 267087. The second author thanks Zhiqin Lu for introducing him this prob-lem and stimulating discussions at Fudan University. The third author thanks Daniel Lenz forgenerously sharing his knowledge on intrinsic metrics and acknowledges the financial support ofthe German Science Foundation (DFG), the Golda Meir Fellowship, the Israel Science Founda-tion (grant No. 1105/10 and No. 225/10) and BSF grant No. 2010214.

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Math. 21 (6) (2009) 1067–1080.

CHAPTER 13

M. Keller, The essential spectrum of the Laplacianon rapidly branching tessellations, Mathematische

Annalen 346 (2010), 51–66.

293

Math. Ann.DOI 10.1007/s00208-009-0384-y Mathematische Annalen

The essential spectrum of the Laplacian on rapidlybranching tessellations

Matthias Keller

Received: 29 January 2009 / Revised: 10 April 2009© Springer-Verlag 2009

Abstract In this paper, we characterize absence of the essential spectrum of theLaplacian under a hyperbolicity assumption for general graphs. Moreover, we pres-ent a characterization for absence of the essential spectrum for planar tessellations interms of curvature.

1 Introduction and main results

The paper is dedicated to investigate the essential spectrum of the Laplacian on graphs.More precisely the purpose is threefold. First, we give a comparison theorem for theessential spectra of the Laplacian ∆ used in the Mathematical Physics community(see for instance [1–4,8,17,18,20,23,24,29]) and the geometric Laplacian ˜∆ used inSpectral Geometry (see for instance [13,14,16,27]) on general graphs.

Second, we consider graphs which are rapidly branching, i.e. the vertex degree isgrowing uniformly as the vertex tends to infinity. We establish a criterion under whichabsence of essential spectrum of the Laplacian ∆ is completely characterized. Thiscriterion will be positivity of a Cheeger constant at infinity introduced in [16], basedon [10,11]. It turns out that in the case of planar tessellating graphs this positivity willbe implied automatically by uniform growth of vertex degree. Moreover, we can inter-pret the rapidly branching property as a uniform decrease of curvature. An immediateconsequence is that these operators have no continuous spectrum.

The third purpose is to demonstrate that∆ and ˜∆may show a very different spectralbehavior. Therefore we discuss a particular class of rapidly branching graphs. Thisdiscussion will also prove independence of our assumptions in the results mentionedabove. In the following introduction we will give an overview. We refer to Sect. 2 forprecise definitions.

M. Keller (B)Mathematical Institute, FSU Jena, 07743 Jena, Germanye-mail: [email protected]

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There is a result of Donnelly and Li [15] on negatively curved manifolds. It showsthat the Laplacian∆ on a rapidly curving manifold has a compact resolvent, i.e. emptyessential spectrum.

Theorem [Donnelly, Li] Let M be a complete, simply connected, negatively curvedRiemannian manifold and K (r) = supK (x, π) | d(p, x) ≥ r the sectional curva-ture for r ≥ 0, where d is the distance function on the manifold, p ∈ M and π is atwo plane in Tx M. If limr→∞ K (r) = −∞, then ∆ on M has no essential spectrumi.e. σess(∆) = ∅.

A remarkable result of Fujiwara [16] provides an analogue in the graph case forthe geometric Laplacian ˜∆.

Theorem (Fujiwara) Let G = (V, E) be an infinite graph. Then σess(˜∆) = 1 if andonly if α∞ = 1.

Here α∞ is a Cheeger constant at infinity. Since the geometric Laplacian ˜∆ is abounded operator the essential spectrum can not be empty. Yet it shrinks to one pointfor α∞ = 1.

We will show that an analogous result holds for the Laplacian ∆, which is usedby mathematical physicists. Let G = (V, E) be an infinite graph. For finite K ⊂ Vdenote by K c its complement V \K and let

mK = infdeg(v) | v ∈ K c and MK = supdeg(v) | v ∈ K c,

where deg : V → N is called the vertex degree, which is defined as the number ofedges emanating from a vertex. Denote

m∞ = limK→∞ mK and M∞ = lim

K→∞ MK .

In the next section, we will be precise about what we mean by the limits. We call agraph rapidly branching if m∞ = ∞. We will prove the following theorems.

Theorem 1 Let G be infinite. For all λ ∈ σess(∆)

m∞ inf σess(˜∆) ≤ λ ≤ M∞ sup σess(˜∆)

and

inf σess(∆) ≤ minm∞,M∞ inf σess(˜∆).

In the first statement we have the convention that if inf σess(˜∆) = 0 and m∞ = ∞ weset m∞ inf σess(˜∆) = 0. The first part of the theorem shows that the essential spectraof the operators ˜∆ and ∆ are related in terms of the minimal and maximal vertexdegree at infinity. The second part gives two options to estimate the infimum of theessential spectrum of ∆ from above.

Our second theorem is the characterization of absence of the essential spectrum.

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The essential spectrum of the Laplacian

Theorem 2 Let G = (V, E) be infinite and α∞ > 0. Then σess(∆) = ∅ if and only ifm∞ = ∞.

Note that m∞ = ∞ does not imply α∞ > 0 or σess(∆) = ∅. This will be discussed inSect. 5.

We may interpret α∞ > 0 as an assumption on the graph to be hyperbolic atinfinity. (See discussion in [22] and the references [19,21,26] found there.) Moreoverthe growth of the vertex degree can be interpreted as the decrease of the curvature.In this way we may understand Theorem 2 as an analogue of a result of Donnellyand Li [15] for ∆ on graphs. For tessellating graphs this analogy will be even moreobvious. There is a recent result of Wojciechowski [29] which proves under strongerassumptions the direction that uniform growth of the vertex degree implies absence ofessential spectrum. Since the continuous spectrum of an operator is always containedin the essential spectrum there is an immediate corollary.

Corollary 1 Let G = (V, E) be infinite, α∞ > 0 and m∞ = ∞. Then ∆ has purepoint spectrum.

The class of examples for which [16] shows absence of essential spectrum are rap-idly branching trees. We will show that the result is also valid for rapidly branchingtessellations. We will formulate the statement in terms of the curvature because thismakes the analogy to Donnelly and Li’s result more obvious. For this sake we definethe combinatorial curvature function κ : V → R for a vertex v ∈ V as it is found in[6,7,22,28] by

κ(v) = 1 − deg(v)

2+

∑

f ∈F,v∈ f

1

deg( f ),

where the face degree deg( f ) denotes the number of vertices contained in a facef ∈ F . For finite K ⊂ V let

κK = supκ(v) | v ∈ K c

and κ∞ = limK→∞ κK . Obviously κ∞ = −∞ is equivalent to m∞ = ∞. Here is ourmain theorem.

Theorem 3 Let G be a tessellation. Then σess(∆) = ∅ if and only if κ∞ = −∞.Moreover κ∞ = −∞ implies σess(˜∆) = 1.The theorem is a special case of Theorem 2. The hyperbolicity assumption α∞ > 0follows from κ∞ = −∞ in the case of tessellating graphs. In particular it is even truethat α∞ = 1 whenever the curvature tends uniformly to −∞.

Klassert et al. [24] proved the absence of finitely supported eigenfunctions for tes-sellations which are non-positively curved in each corner. To assume this in our caseis natural, since non-positive corner curvature is already implied by a vertex degreegreater or equal 6. So under this assumption we have pure point spectrum such thatall eigenfunctions are supported on an infinite set of vertices.

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The paper is structured as follows. In Sect. 2, we will define the versions of theLaplacian which appear in different contexts of the literature. We discuss Fujiwara’sTheorem which can be understood as a result on compact operators. In Sect. 3 weprove Theorem 1 and 2. In Sect. 4, we give an estimate of the Cheeger constant atinfinity for planar tessellations and prove Theorem 3. Finally in Sect. 5, we discussa class of examples which shows that for general graphs ∆ and ˜∆ can have a quitedifferent spectral behavior.

2 The geometric Laplacian ˜∆ in terms of compact operators

Let G = (V, E) be a connected graph with finite vertex degree at each vertex. For apositive weight function g : V → R+ let

2(V, g) =

ϕ : V → R | 〈ϕ, ϕ〉g =∑

v∈V

g(v)|ϕ(v)|2 < ∞

,

cc(V ) = ϕ : V → R | |supp ϕ| < ∞

where supp is the support of a function. For g = 1 we write 2(V ). Notice that 2(V, g)is the completion of cc(V ) under 〈·, ·〉g . For g = deg it is clear that 2(V, deg) ⊆ 2(V )and if supv∈V deg(v) < ∞ then 2(V ) = 2(V, deg).

Define the operators A and D on cc(V ) by

(Aϕ)(v) =∑

u∼vϕ(u) and (Dϕ)(v) = deg(v)ϕ(v).

The operator A is often called the adjacency matrix. Since we assumed that the graphhas no isolated vertices the operator D has a bounded inverse.

The Laplace operator plays an important role in different areas of mathematics.Yet there occur different versions of it. To avoid confusion we want to discuss thembriefly. We start with the Laplacian used in the Mathematical Physicist community inthe context of Schrödinger operators. For reference see e.g. [8,11] (and the referencestherein) or in more recent publications like [1–4,12,17,18,20,23,24].

(1) The operator D − A defined on cc(V ) yields the following form

〈dϕ, dϕ〉 = 1

2

∑

v∈V

∑

u∼v|ϕ(u)− ϕ(v)|2.

The unique self adjoint extension on a subspace D(∆) of 2(V ) of the operator cor-responding to this form will be denoted by ∆. It gives for ϕ ∈ D(∆) and v ∈ V

(∆ϕ) (v) = deg(v)ϕ(v)−∑

u∼vϕ(u). (1)

Notice that ∆ is unbounded if there is no bound on the vertex degree.

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We next introduce the geometric Laplacian. Two unitarily equivalent versions arefound in the literature. They are given as follows.

(2) Let

˜∆ = I − Ã = I − D−1 A

be defined on 2(V, deg), where I is the identity operator. It is easy to see that ˜∆ isbounded and self adjoint. For ϕ ∈ 2(V, deg) and v ∈ V it gives

(˜∆ϕ)(v) = ϕ(v)− 1

deg(v)

∑

u∼vϕ(u).

The matrix Ã is often called the transition matrix. This version of the geometricLaplacian can be found for instance in [13,14,16,27] and many others. Moreover ˜∆

is the self adjoint extension on 2(V, deg) of the operator associated with the formgiven by (1) on cc(V ).

(3) There is a unitarily equivalent version as discussed e.g. in [9]. Let

∆ = I − A = I − D− 12 AD− 1

2

be defined on 2(V ). It gives for ϕ ∈ 2(V ) and v ∈ V

(∆ϕ)(v) = ϕ(v)−∑

u∼v

1√

deg(u) deg(v)ϕ(u).

Notice that the operator

D121,deg : 2(V, deg) → 2(V ), ϕ → √

deg · ϕ,

is an isometric isomorphism and we denote its inverse by D− 1

2deg,1. Then

∆ = D121,deg

˜∆D− 1

2deg,1.

Moreover on cc(V )

∆ = D12 ∆D

12 .

Furthermore, we define the Dirichlet restrictions of these operators. For a set K ⊆ Vlet PK : 2(V, g) → 2(K c, g) be the canonical projection and iK : 2(K c, g) →2(V, g) its dual operator, which is the continuation by 0 on K . For an operator B on2(V, g) we write

BK = PK BiK .

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Hence we can speak of∆K , ˜∆K or ∆K on K c with Dirichlet boundary conditions. Inmost cases K will be a finite set.

For a graph G and finite K ⊆ V define the Cheeger constant, see [10,14,16],

αK = infW⊆K c, |W |<∞

|∂E W |A(W )

,

where |W | denotes the cardinality of a set W , ∂E W is the set of edges which have onevertex in W and one outside and A(W ) = ∑

v∈W deg(v). Let W ⊆ K c, for K finiteand χ the characteristic function of W . Two simple calculations, mentioned in [13]yield

⟨

˜∆Kχ, χ⟩

deg = 〈∆Kχ, χ〉 = |∂E W |

and

〈χ, χ〉deg = 〈Dχ, χ〉 = A(W ).

This gives

αK = infW⊆K c, |W |<∞

⟨

˜∆χWχW⟩

deg

〈χW , χW 〉deg. (2)

The set K (V ) of finite subsets of V is a net under the partial order ⊆. We say thata function F : K (V ) → R, K → FK converges to F∞ ∈ R if for all ε > 0there is a Kε ∈ K (V ) such that |FK − F∞| < ε for all K ⊇ Kε . We then writelimK→∞ FK = F∞. With this convention we define the Cheeger constant at infinityas in [16] by

α∞ = limK→∞αK .

The limit always exists since αK ≤ αL ≤ 1 for finite K ⊆ L ⊆ V . Therefore we canthink of taking the limit over distance balls of an arbitrary vertex.

The next part is dedicated to a discussion of [16]. We will look at the result from theperspective of compact operators. The proof is based on two propositions which holdfor general graphs. We present them here as norm estimates on the transition matrix.As shown in [13] inf σ(˜∆) ≤ α. This easily leads to the following statement.

Proposition 1 For any finite set K ⊆ V

∥

∥ÃK∥

∥ ≥ 1 − αK .

In particular, inf σ(˜∆K ) ≤ αK .

The second proposition is derived from Proposition 1 in [16] which is inspired bya calculation in [14].

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Proposition 2 For any finite set K ⊆ V

∥

∥ÃK∥

∥ ≤√

1 − α2K .

The next theorem is direct consequence of the two propositions above and can befound in [16].

Theorem 4 For K ⊆ V finite

1 −√

1 − α2K ≤ ˜∆K ≤ 1 +

√

1 − α2K .

The essential parts (ii) ⇔ (v), (ii) ⇒ (iii) of the next theorem are already found in[16]. The remaining statements are minor extensions. Knowing Proposition 1 and 2one only needs standard methods for compact operators (and Propositions 3 in Sect. 3)to prove the following theorem.

Theorem 5 Let G be infinite. The following are equivalent.

(i) σess(˜∆) consists of one point.

(ii) σess(˜∆) = 1.(iii) Ã is compact.

(iv) limK→∞ ‖ÃK ‖ = 0.

(v) α∞ = 1.

3 The essential spectrum of ∆

In this section, we compare the operators ˜∆ and ∆. We will establish bounds on theessential spectrum of ∆ by using bounds obtained for ˜∆. Therefore, we will firstlystate a well known proposition which allows us to determine the essential spectrumof an operator via its restriction on the complement of larger and larger sets. Thenwe prove two propositions which estimate the infimum of the essential spectrum of∆ from below and above. These will be the ingredients for the proofs of Theorem 1and 2.

Proposition 3 Let G = (V, E) be infinite and B a self adjoint operator which isbounded from below and cc(V ) dense in D(B) ⊆ 2(V, g) w.r.t. to the graph norm.Then

inf σess(B) = limK→∞ inf

ϕ ∈ cc(V )supp ϕ ⊆ K c

〈Bϕ, ϕ〉g

〈ϕ, ϕ〉g= lim

K→∞ inf σ(BK ),

sup σess(B) ≤ limK→∞ sup

ϕ ∈ cc(V )supp ϕ ⊆ K c

〈Bϕ, ϕ〉g

〈ϕ, ϕ〉g= lim

K→∞ sup σ(BK ).

If B is bounded, we have equality in the second formula.

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The proof follows by standard techniques, that is why we omit it here. (Note that oneuses limK→∞

∥

∥E]−∞,λ]iK∥

∥ = 0 for λ < inf σess(B), where E]−∞,λ] is the spectralprojection of B.)

Since ˜∆ and ∆ are unitarily equivalent it makes no difference whether we comparethe operators ˜∆ and∆ or the operators ∆ and∆. Yet∆ and ∆ are defined on the sameHilbert space, so it seems to be notationally easier to compare them. However to dothis the following identity is vital. For ϕ ∈ cc(K c) one can calculate

〈∆Kϕ, ϕ〉〈ϕ, ϕ〉 = 〈D

12K

∆K D12Kϕ, ϕ〉

〈ϕ, ϕ〉 = 〈 ∆K D12Kϕ, D

12Kϕ〉

〈D12Kϕ, D

12Kϕ〉

〈DKϕ, ϕ〉〈ϕ, ϕ〉 . (3)

Proposition 4 Let G be an infinite graph. Then for λ ∈ σess(∆)

m∞ inf σess(∆) ≤ λ ≤ M∞ sup σess(∆).

Proof Let K ⊂ V be finite. By Eq. 3 we have for ϕ ∈ cc(K c)

〈∆Kϕ, ϕ〉〈ϕ, ϕ〉 ≥ 〈 ∆K D

12Kϕ, D

12Kϕ〉

〈D12Kϕ, D

12Kϕ〉

infv∈supp ϕ

deg(v).

For every ψ ∈ cc(K c) there is an ϕ ∈ cc(K c) such that ψ = D12Kϕ. Furthermore

cc(K c) is dense in the domain of ∆K which allows us to conclude

inf σess(∆) = inf σess(∆K ) ≥ m∞ inf σ(∆K ).

By Proposition 3 this yields the lower bound. If M∞ = ∞ the upper bound is infinity.Otherwise by Eq. 3 sup σ(∆K ) ≤ M∞ sup σ(∆K ) and again by Proposition 3 we havethe upper bound. Proposition 5 Let G be an infinite graph. Then

inf σess(∆) ≤ minm∞,M∞ inf σess(∆).

Proof Let vn ∈ V , n ∈ N be pairwise distinct such that deg(vn) ≤ m∞. Moreover letχn the characteristic function of vn . For K finite such that vn ∈ K c we have

infϕ∈cc(K c)

〈∆Kϕ, ϕ〉〈ϕ, ϕ〉 ≤ 〈∆Kχn, χn〉 = deg(vn) ≤ m∞.

By Proposition 3 we have inf σess(∆) ≤ m∞. On the other hand we have by Eq. 3inf σ(∆K )≤ MK inf σ(∆K ). By Proposition 3 we get inf σess(∆)≤ M∞ inf σess(∆).

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Proof of Theorem 1 Recall that the operators ˜∆ and ∆ are unitarily equivalent. Thusσess(˜∆) = σess(∆). The Theorem follows from Propositions 4 and 5. Proof of Theorem 2 By Theorem 4 we have

inf σ(∆K ) ≥ 1 −√

1 − α2K ≥ 0.

Thus by taking the limits Proposition 3 yields inf σess(∆) > 0 if α∞ > 0. Proposi-tions 4 and 5 give the desired result.

4 Rapidly branching tessellations

Examples discussed in [16] are rapidly branching trees. Fujiwara showed that for treesα∞ = 1 is implied by m∞ = ∞. Therefore by Theorem 2 we have σess(∆) = ∅ inthe case of trees. In this section, we want to extend the class of examples to tessella-tions. We do this by showing that α∞ = 1 is implied by m∞ = ∞ for tessellationsas well. For planar graphs tessellations are quite well understood. We only recall thedefinitions and refer the reader to [6,7] and the references contained therein.

Let G = (V, E) be a planar, locally finite graph without loops and multipleedges, embedded in R2. We denote the set of closures of the connected componentsin R2\⋃

e∈E e by F and refer to the elements of F as faces of G. We may writeG = (V, E, F). A union of faces is called a polygon if it is homeomorphic to a closeddisc in R2 and its boundary is a closed path of edges without repeated vertices. Thegraph G is called a tessellation or tessellating if the following conditions are fulfilled.

i) Any edge is contained in precisely two different faces.ii) Two faces are either disjoint or intersect in a unique edge or vertex.

iii) All faces are polygons.

Note that a planar tessellating graph is always infinite. From now on let G = (V, E, F)be tessellating and we do not distinguish between the graph and its embedding in R2.Recall that the vertex degree deg(v) is the number of edges emanating from a vertexv and the face degree deg( f ) is the number of vertices contained in a face f . For a setW ⊆ V let GW = (W, EW , FW ) be the induced subgraph, which is the graph withvertex set W and the edges of E which have two vertices in W . Euler’s formula statesfor a connected finite subgraph GW

|W | − |EW | + |FW | = 2. (4)

Observe that the 2 on the right hand side occurs since we also count the unbounded face.Euler’s formula is a part of mathematical folklore. A proof can be found for instancein [5]. We denote by ∂F W the set of faces in F which contain an edge of ∂E W . Infact each face in ∂F W contains at least two edges in ∂E W . Therefore |∂F W | ≤ |∂E W |follows easily. Moreover we define for finite W ⊆ V the inner degree of a face f ∈ Fby

degW ( f ) = |v ∈ W | v ∈ f |

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M. Keller

Finally let C(W ) be the number of connected components in GV \W . Loosely speakingit is the number of holes in GW .

We need two important formulas which hold for arbitrary finite subgraphs GW =(W, EW , FW ) of G. Recall A(W ) = ∑

v∈W deg(v). The first formula can be easilyrechecked. It reads

A(W ) = 2|EW | + |∂E W |. (5)

As for the second formula note that FW has faces which are not in F . Nevertheless

|FW | − C(W ) = |FW ∩ F |.

This is the number of bounded faces which are enclosed by edges of EW . Thus sortingthe following sum over vertices according to faces gives the second formula

∑

v∈W

∑

f ∈F, f v

1

deg( f )= |FW | − C(W )+

∑

f ∈∂F W

degW ( f )

deg( f ). (6)

Lemma 1 Let G = (V, E, F) be a tessellating graph. Then for a finite and connectedset W ⊆ V

|∂E W | ≥ A(W )− 6(|W | + C(W )− 2).

Proof By the tessellating property we have

∑

f ∈∂F W

degW ( f ) ≥ |∂E W |.

Moreover deg( f ) ≥ 3 for f ∈ F . Combining this with Eq. 6 we obtain

|FW | ≤ 1

3

⎛

⎝

∑

v∈W

∑

f v1 −

∑

f ∈∂F W

degW ( f )

⎞

⎠ + C(W )

≤ 1

3(A(W )− |∂E W |)+ C(W ).

Applying Euler’s formula (4) and equation (5) to the left hand side of the inequalityabove, we obtain

2 ≤ |W | − 1

6(A(W )− |∂E W |)+ C(W ),

which yields the Lemma. Now we give an estimate from below of the Cheeger constant at infinity.

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Proposition 6 Let G = (V, E, F) be a tessellating graph. Then

α∞ ≥ 1 − limK→∞ sup

v∈K c

6

deg(v).

Proof We assume w.l.o.g. that the finite sets K are distance balls. To calculate αK wecan restrict ourselves to finite sets W ⊂ V , which are connected. Otherwise we find aconnected component W0 of W such that |∂E W0|/A(W0) ≤ |∂E W |/A(W ). Moreoverwe can choose W such that C(W ) ≤ 2. Otherwise we find a superset W1 of W suchthat |∂E W1|/A(W1) ≤ |∂E W |/A(W ).

Obviously A(W ) ≥ |W | infv∈W deg(v). By Lemma 1

|∂E W |A(W )

≥ A(W )− 6|W |A(W )

≥ 1 − 6

infv∈W deg(v).

Hence we have αK ≥ 1 − supv∈K c 6/ deg(v). We obtain the result by taking the limitover all finite sets. Remark 1 The relation between curvature and the Cheeger constant can be presentedin more detail than we need it for our purpose here. See therefore [25].

Proof of Theorem 3 Let κ∞ = −∞. This is obviously equivalent to m∞ = ∞ whichimplies α∞ = 1 by Proposition 6. Thus by Theorems 5 and 1 we obtain σess(˜∆) = 1and σess(∆) = ∅. On the other hand Proposition 5 tells us that σess(∆) = ∅ impliesm∞ = ∞ and thus κ∞ = −∞. Remark 2 The implication that σess(∆) = ∅ follows from κ∞ = −∞ can also beobtained in an alternative way. Higuchi [22] and Woess [28] showed independentlythat αK > 0 whenever κK < 0 for K = ∅. Since m∞ = ∞ is implied by κ∞ = −∞we can apply Theorem 1 immediately.

5 A further class of rapidly branching graphs

In this section, we want to discuss a class of examples which demonstrates that∆ and˜∆ can show very different spectral phenomena. In particular these examples prove theindependence of our assumptions in Theorem 2.

Let G(n) = (V (n), E(n)) be the full graph with n vertices. For γ ≥ 0 and c ≥ 1let

Nγ,c : N → N, n → n[cnγ ],

where [x] is the the smallest integer bigger than x ∈ R. Denote N1 = 1, N2 =max[c], 2 and for k ≥ 3

Nk = Nγ,c(Nk−1).

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M. Keller

We construct the graph Gγ,c as follows. We start by connecting the vertex in G(N1)

with each vertex in G(N2). We proceed by connecting each vertex in G(Nk) uniquelywith [cN γ

k ] vertices in G(Nk+1) for k ∈ N. Obviously Gγ,c is rapidly branchingwhenever γ > 0 or c > 1, in fact Nk ≥ 2k−1. From another point of view Gγ,c isa ‘tree’ of branching number [cN γ

k ] in the kth generation, where we connected thevertices of each generation with one another. The next theorem shows a scheme of thequite different behavior of the sets σess(∆) and σess(˜∆) for the graphs Gγ,c.

Theorem 6 For γ ≥ 0 and c ≥ 1 let Gγ,c be as above.

(1) If γ = 0, then α∞ = 0, inf σess(˜∆) = 0 and inf σess(∆) ≤ [c].(2) If γ ∈ ]0, 1[, then α∞ = 0, inf σess(˜∆) = 0 and σess(∆) = ∅.(3) If γ = 1, then α∞ = c

1+c , inf σess(˜∆) ∈ ]0, 1[ and σess(∆) = ∅.

(4) If γ > 1, then α∞ = 1, σess(˜∆) = 1 and σess(∆) = ∅.

As mentioned above all graphs Gγ,c are rapidly branching if γ > 0 or c > 1. Thetheorem shows the independence of our assumptions and thus optimality of the result.More precisely the case γ = 0 shows that m∞ = ∞ alone does not imply σess(∆) = ∅.On the other hand the case γ ∈ ]0, 1[ makes clear that σess(∆) = ∅ does not implyα∞ > 0. Moreover when γ = 1 we see that m∞ = ∞ and α∞ > 0 does not implyα∞ = 1. The last case is an example where σess(˜∆) = 1 and σess(∆) = ∅ as in thecase of trees and tessellations.

For a graph G denote by Bn the set of vertices which have distance n ∈ N or lessfrom a fixed vertex v0 ∈ V . In our context choose v0 as the unique vertex in G(N1).

The intuition behind the theorem is as follows. Let Sn,k = Bn\Bk , n > k andχ = χSn,k its characteristic function. Then one can calculate

⟨

∆Bkχ, χ⟩

〈χ, χ〉 = |∂E Sn,k |A(Sn,k)

A(Sn,k)

|Sn,k | ∼ c

N 1−γn + c

N γn

(

N 1−γn + c

)

= cN γn .

The left hand side may thought to be close to inf σ(∆Bk ). Moreover the first factorafter the equality sign may thought to be close to αBk . If this is true we can control thegrowth and the decrease of these terms by γ . For instance inf σ(∆Bk ) would increaseto infinity although αBk tends to zero for γ < 1.

We denote for a vertex v ∈ Bk

deg±(v) = |w ∈ Sk±1 | v ∼ w|,

where we set Sk = Bk\Bk−1 for k ≥ 2. To prove the theorem we will need the fol-lowing three Lemmata. Lemma 1.5 of [13] and the remark that follows it yield thefollowing.

Lemma 2 Let G be a graph. If (deg+(v) − deg−(v))/ deg(v) ≥ C for all v ∈ Bcn

then αBn ≥ C.

With the help of this Lemma we will prove the statements for α∞ on the respectivegraphs.

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Lemma 3 Let c ≥ 1.

1. If γ < 1 then α∞ = 0.2. If γ = 1 then α∞ = c

1+c .3. If γ > 1 then α∞ = 1.

Proof We get an estimate from above by calculating

αBn−1 ≤ |∂E Sn|A(Sn)

= Nn[

cN γn] + Nn

Nn(Nn − 1)+ Nn[

cN γn] + Nn

= Nn[

cN γn] + Nn

N 2n + Nn

[

cN γn] .

To obtain a lower bound for αBn−1 we use Lemma 2 and calculate

infv∈Bc

n−1

deg+(v)− deg−(v)deg(v)

= infk≥n

[

cN γ

k

] − 1

1 + Nk + [

cN γ

k

] =[

cN γn] − 1

1 + Nn + [

cN γn] .

One gets the desired result by letting n tend to infinity.

The next lemma is crucial to show absence of essential spectrum for∆when γ > 0.

Lemma 4 Let γ > 0 and ϕk functions in cc(Bck−1) such that ‖ϕk‖ ≤ 1 and

⟨

∆Bk−1ϕk, ϕk⟩ ≤ C for all k ∈ N and some constant C > 0. Then

limk→∞ ‖ϕk‖ = 0.

Proof Choose ϕk , k ∈ N as assumed. Denote by ϕ(i)k the restriction of ϕk to Si =Bi\Bi−1 for i ≥ k and choose m > k such that supp ϕk ⊆ Bm . Then an estimate onthe form of ∆Bk−1 reads

⟨

∆Bk−1ϕk, ϕk⟩ ≥

m∑

i=k

∑

v∈Si

∑

w∈Si+1,w∼v|ϕk(v)− ϕk(w)|2

≥m

∑

i=k

⎛

⎝

∑

v∈Si

[

cN γ

i

]

ϕ2k (v)+

∑

w∈Si+1

ϕ2k (w)

−2∑

v∈Si

∑

w∈Si+1,w∼vϕk(v)ϕk(w)

⎞

⎠

123

M. Keller

≥m

∑

i=k

⎛

⎜

⎝

[

cN γ

i

]∑

v∈Si

ϕ2k (v)+

∑

w∈Si+1

ϕ2k (w)

−2

⎛

⎝

[

cN γ

i

]∑

v∈Si

ϕ2k (v)

⎞

⎠

12⎛

⎝

∑

w∈Si+1

ϕ2k (w)

⎞

⎠

12⎞

⎟

⎠

=m

∑

i=k

(

[

cN γ

i

]12

∥

∥

∥ϕ(i)k

∥

∥

∥ −∥

∥

∥ϕ(i+1)k

∥

∥

∥

)2

In the second step we used that each vertex in Si is uniquely adjacent to [cN γ

i ] verticesin Si+1 for k ≤ i ≤ m and in the third step we used the Cauchy–Schwarz inequality.We assumed

⟨

∆Bk−1ϕk, ϕk⟩ ≤ C and in particular this is true for every term in sum we

estimated above. Moreover∥

∥

∥ϕ(i+1)k

∥

∥

∥ ≤ ‖ϕk‖ ≤ 1 for k ≤ i ≤ m and thus

∥

∥

∥ϕ(i)k

∥

∥

∥ ≤√

C +∥

∥

∥ϕ(i+1)k

∥

∥

∥

c12 N

γ2

i

≤√

C + 1

c12 N

γ2

i

.

Set C0 = (√

C + 1)/c12 . Since the sequence

(

N− γ

2i

)

is summable we deduce

‖ϕk‖ ≤m

∑

i=k

∥

∥

∥ϕ(i)k

∥

∥

∥ ≤ C0

m∑

i=k

N− γ

2i ≤ C0

∞∑

i=k

N− γ

2i < ∞.

We now let k tend to infinity and conclude limk→∞ ‖ϕk‖ = 0. Proof of Theorem 6 From Propositions 1, 2 and 3 we can deduce

1 −√

1 − α2∞ ≤ inf σess(˜∆) ≤ α∞.

Thus by Lemma 3 we get the assertion for α∞ and inf σess(˜∆).If γ = 0 we get for the characteristic function χ = χSn,k of Sn,k = Bn\Bk , n > k

⟨

∆Bkχ, χ⟩

〈χ, χ〉 = [c](Nk + Nn)∑n

i=k+1 Ni=

[c](

NkNn

+ 1)

1 + ∑n−1i=k+1

NiNn

.

Hence by taking the limit over n we have by Proposition 3 that inf σess(∆) ≤ [c].Let γ > 0 and let ϕk be functions in cc(Bc

k+1) such that ‖ϕk‖ = 1 and

limk→∞

⟨

∆Bk+1ϕk, ϕk⟩ = inf σess(∆).

123


This is possible by Proposition 3 and a diagonal sequence argument. As ‖ϕk‖ = 1 byLemma 4 the term

⟨

∆Bkϕk, ϕk⟩

tends to infinity. Thus the essential spectrum of ∆ isempty. Acknowledgments I take this chance to express my gratitude to Daniel Lenz for all the fruitful discus-sions, helpful suggestions and his guidance during this work. I also like to thank Norbert Peyerimhoff forall helpful hints during my visit in Durham. This work was partially supported by the German ResearchCouncil (DFG) and partially by the German Business Foundation (sdw).

References

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2. Aizenman, M., Sims, R., Warzel, S.: Stability of the absolutely continuous spectrum of randomSchrödinger operators on tree graphs. Prob. Theory Relat. Fields 136, 363–394 (2006)

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4. Breuer, J.: Singular continuous spectrum for the Laplacian on certain sparse trees. Commun. Math.Phys. 269(3), 851–857 (2007)

5. Bollobás, B.: Graph Theory: An Introductory Course. Springer-Verlag (1979)6. Baues, O., Peyerimhoff, N.: Curvature and geometry of tessellating plane graphs. Discrete Comput.

Geom. 25 (2001)7. Baues, O., Peyerimhoff, N.: Geodesics in non-positively curved plane tessellations. Adv. Geom.

6(2), 243–263 (2006)8. Cycon, H.L., Froese, R.G., Kirsch, W., Simon, B.: Schrödinger Operators, Springer Verlag (1987)9. Chung, F.: Spectral graph theory. CBMS Regional Conference Series in Mathematics, No. 92, Amer-

ican Mathematical Society, New York (1997)10. Cheeger, J.: A lower bound for the lowest eigenvalue of the Laplacian. In: Problems in Analysis.

A Symposium in honor of S. Bochner, pp. 195–199. Princeton University Press, Princeton (1970)11. Dodziuk, J.: Difference equations, isoperimetric inequalities and transience of certain random walks.

Trans. Am. Math. Soc. 284, 787–794 (1984)12. Dodziuk, J.: Elliptic operators on infinite graphs. In: Proceedings of the Conference ’Krzysztof

Wojciechowski 50 years—Analysis and Geometry of Boundary Value Problems’, Roskilde, Denmark(to appear). http://xxx.lanl.gov/abs/math.SP/0509193

13. Dodziuk, J., Karp, L.: Spectral and function theory for combinatorial Laplacians. In: Durrett, R.,Pinsky, M.A. (eds.) Geometry of Random Motion, vol. 73, pp. 25–40. AMS Contemporary Mathe-matics, New York (1988)

14. Dodziuk, J., Kendall, W.S.: Combinatorial Laplacians and isoperimetric inequality. In: Elworthy, K.D.(ed.) From Local Times to Global Geometry, Control and Physics. Longman Scientific and Technical,pp. 68-75 (1986)

15. Donnelly, H., Li, P.: Pure point spectrum and negative curvature for noncompact manifolds. DukeMath J. 46, 497–503 (1979)

16. Fujiwara, K.: Laplacians on rapidly branching trees. Duke Math. J. 83(1), 191–202 (1996)17. Froese, R., Hasler, D., Spitzer, W.: Transfer matrices, hyperbolic geometry and absolutely continuous

spectrum for some discrete Schrödinger operators on graphs. J. Funct. Anal. 230, 184–221 (2006)18. Georgescu, V., Golénia, S.: Isometries, fock spaces and spectral analysis of Schrödinger operators on

trees. J. Funct. Anal. 227, 389–429 (2005)19. Ghys, E., de la Harpe, P. (ed.): Sur les groupes hyperboliques d’après Mikhael Gromov. Progress in

Mathematics 83, Birkhäuser, Basel (1990)20. Golénia, S.: C*-algebra of anisotropic Schrödinger operators on trees. Annales Henri Poincaré

5(6), 1097–1115 (2004)21. Gromov, M.: Hyperbolic groups. In: Gersten, S.M. (ed.) Essays in group theory, pp. 75–263. M.S.R.I.

Publ. 8, Springer, Heidelberg (1987)22. Higuchi, Y.: Combinatorial curvature for planar graphs. J. Graph Theory (2001)

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23. Klein, A.: Absolutely continuous spectrum in the Anderson model on the Bethe lattice. Math. Res.Lett. 1, 399–407 (1994)

24. Klassert, S., Lenz, D., Peyerimhoff, N., Stollmann, P.: Elliptic operators on planar graphs: uniquecontinuation for eigenfunctions and nonpositive curvature. Proc. AMS 134, No. 5 (2005)

25. Keller, M., Peyerimhoff, N.: Cheeger constants, growth and spectrum of locally tessellating planargraphs. arXiv:0903.4793

26. Lyndon, R.S., Schupp, P.E.: Combinatorial group theory, Springer (1977)27. Woess, W.: Random Walks on Infinite Graphs and Groups. Cambridge Tracts in Mathematics 138.

Cambridge University Press (2000)28. Woess, W.: A note on tilings and strong isoperimetric inequality. Math. Proc. Camb. Philos. Soc. 124,

385–393 (1998)29. Wojciechowski, R.: Heat kernel and essential spectrum of infinite graphs. Indiana Univ. Math. J. (to

appear)

123

CHAPTER 14

M. Keller, N. Peyerimhoff, Cheeger constants,growth and spectrum of locally tessellating planar

graphs, Mathematische Zeitschrift 268 (2011),871–886.

311

Math. Z. (2011) 268:871–886DOI 10.1007/s00209-010-0699-0 Mathematische Zeitschrift

Cheeger constants, growth and spectrum of locallytessellating planar graphs

Matthias Keller · Norbert Peyerimhoff

Received: 28 August 2009 / Accepted: 8 February 2010 / Published online: 20 March 2010© Springer-Verlag 2010

Abstract In this article, we study relations between the local geometry of planar graphs(combinatorial curvature) and global geometric invariants, namely the Cheeger constantsand the exponential growth. We also discuss spectral applications.

1 Introduction

A locally tessellating planar graph G is a tiling of the plane with all faces to be polygons withfinitely or infinitely many boundary edges. The edges of G are continuous rectifiable curveswithout self-intersections. Faces with infinitely many boundary edges are called infinigonsand occur, e.g., in the case of planar trees. The sets of vertices, edges and faces of G aredenoted by V, E and F (see the beginning of Sect. 2 for precise definitions). The functiond(v,w) denotes the combinatorial distance between two vertices v,w ∈ V , where each edgeis assumed to have combinatorial length one. For any pair v,w of adjacent vertices we writev ∼ w.

Useful local concepts of a planar graph G are combinatorial curvature notions. The finestcurvature is defined on the corners of G. A corner is a pair (v, f ) ∈ V × F , where v is avertex of the face f . The corner curvature κC is defined as

κC (v, f ) = 1

|v| + 1

| f | − 1

2,

M. Keller (B)Mathematical Institute, Friedrich-Schiller-University Jena,07743 Jena, Germanye-mail: [email protected]

N. PeyerimhoffDepartment of Mathematical Sciences, University of Durham,Durham DH1 3LE, UKe-mail: [email protected]

123

872 M. Keller, N. Peyerimhoff

where |v| and | f | denote the degree of the vertex v and the face f . If f is an infinigon, weset | f | = ∞ and 1/| f | = 0. The curvature at a vertex v ∈ V is given by

κ(v) =∑

f :v∈ f

κC (v, f ) = 1 − |v|2

+∑

f :v∈ f

1

| f | .

For a finite set W ⊂ V we define κ(W ) = ∑v∈W κ(v).

These combinatorial curvature definitions arise naturally from considerations of the Eulercharacteristic and tessellations of closed surfaces, and they allow to prove a combinato-rial Gauß-Bonnet formula (see [2, Theorem 1.4]). Similar combinatorial curvature notionshave been introduced by many other authors, e.g. [14,16,28,31]. Let us mention two globalgeometric consequences of the curvature sign:

• In [5] it was proved that strictly positive vertex curvature implies finiteness of a graph,thus proving a conjecture of Higuchi (which is a discrete analogue of Bonnet–Myerstheorem in Riemannian geometry). This question was investigated before by Stone [28].

• The cut locus Cut(v) of a vertex v consists of all vertices w ∈ V , at which dv := d(v, ·)attains a local maximum, i.e., we have w ∈ Cut(v) if dv(w

′) ≤ dv(w) for all w′ ∼ w.If G is a plane tessellation with non-positive corner curvature, then G is without cutlocus, i.e., we have Cut(v) = ∅ for all v ∈ V . This fact can be considered as a com-binatorial analogue of the Cartan–Hadamard theorem (for a proof and more details see[3, Theorem 1]).

For a finite subset W ⊂ V , let vol(W ) = ∑v∈W |v|. We consider the following two types

of Cheeger constants:

α(G) := infW ⊆ V,

|W | < ∞

|∂E W ||W | and α(G) := inf

W ⊆ V,

|W | < ∞

|∂E W |vol(W )

, (1)

where ∂E W is the set of all edges e ∈ E connecting a vertex in W with a vertex in V \W .The quantity α(G) is called the physical Cheeger constant and α(G) the geometric Cheegerconstant of the graph G. The attributes physical and geometric are motivated by the factthat these constants are closely linked to two types of Laplacians (see, e.g. [21,30]) and thatthe first type is used in the community of Mathematical Physics whereas the second appearsfrequently in the context of Spectral Geometry. Cheeger constants are invariants of the globalasymptotic geometry. They are important geometric tools for spectral considerations (bothin setting of graphs and of Riemannian manifolds) and play a prominent role in the topic ofexpanders and Ramanujan graphs (see [20] for a very recommendable survey on this topic).

Natural model spaces are the (p, q)-regular plane tessellations G p,q : every vertex in G p,q

has degree p and every face has degree q . (In the case 1p + 1

q < 12 , G p,q can be realized

as a regular tessellation of the Poincaré disc model of the hyperbolic plane by translates ofa regular compact polygon.) The graphs G p,q can be considered as discrete counterparts ofconstant curvature space forms in Riemannian Geometry. The Cheeger constants of theseregular graphs are explicitly given:

Theorem (see [15,18,19]) Let 1p + 1

q ≤ 12 . Then

α(G p,q) = p − 2

p

√

1 − 4

(p − 2)(q − 2).

123

Cheeger constants, growth and spectrum of locally tessellating planar graphs 873

Let us now leave the situation of regular tessellations. It is known that the Cheeger constantsof general negatively curved planar graphs are strictly positive (see [6,16,31]). Moreover,for infinite planar graphs G with |v| ≥ p and | f | ≥ q for almost all vertices and faces andc := 1

2 − 1p − 1

q > 0, the following estimate was shown in [26]:

α(G) ≥ 2pqc

3q − 8. (2)

Next we introduce a bit of notation before we state our explicit lower Cheeger constant esti-mates. The variables p, q in this paper always represent a pair of numbers 3 ≤ p, q ≤ ∞satisfying 1

p + 1q ≤ 1

2 (note that we use 1/∞ = 0). For such a pair (p, q), let

C p,q :=

⎧⎪⎨

⎪⎩

1, if q = ∞,

1 + 2q−2 , if q < ∞ and p = ∞,(

1 + 2q−2

) (1 + 2

(p−2)(q−2)−2

), if p, q < ∞.

(3)

Then we have

Theorem 1 (Cheeger constant estimate) Let G = (V, E, F) be a locally tessellating planargraph such that |v| ≤ p for all v ∈ V and | f | ≤ q for all f ∈ F. (Note that p = ∞ orq = ∞ means no condition on the vertex of face degrees.) Let C p,q be defined as in (3).

(a) Assume that C := infv∈V −κ(v) is strictly positive. Then

α(G) ≥ 2C p,qC.

(b) Assume that c := infv∈V − 1|v|κ(v) is strictly positive. Then

α(G) ≥ 2C p,qc.

The above estimates are sharp in the case of regular trees (in which case q = ∞).

The proof of this theorem is given in Sect. 2. Observe that the constant C p,q ≥ 1 in (3)becomes largest if the graph G in Theorem 1 has both finite upper bounds for vertex and facedegrees. (A shorter expression for C p,q is q(p−2)

(p−2)(q−2)−2 , which we have to interpret in theright way if q = ∞ or p = ∞.)

Let us study our estimate in the regular case G = G p,q : In this case our estimate yields

(p − 2)

(1 − 2

(p − 2)(q − 2) − 2

)≤ α(G p,q).

On the other hand, a straightforward calculation leads to the following upper bound

α(G p,q) = (p − 2)

√

1 − 4

(p − 2)(q − 2)≤ (p − 2)

(1 − 2

(p − 2)(q − 2) − 1

),

which shows that our lower bound is very close to the correct value. Mohar’s estimate (2)in this situation coincides with ours in the particular case (p, q) = (∞, 3), and becomesconsiderably weaker for q ≥ 4 or p < ∞.

Remark Every infinite connected graph G = (V, E) with |v| ≤ p has physical Chee-ger constant α(G) ≤ p − 2. To see this, choose an infinite path v0, v1, v2, . . . and letWn := v0, v1, . . . , vn. Then we have

|∂E Wn ||Wn | ≤ 2(p − 1) + (n − 1)(p − 2)

n + 1,

123


which implies

α(G) ≤ limn→∞

|∂E Wn ||Wn | = p − 2.

The same arguments show α(Tp) = p−2 and α(Tp) = p−2p , where Tp denotes the p-regular

infinite tree.

Next, we turn to other global asymptotic invariants related to the growth of an infinitegraph G = (V, E). For a fixed center v0 ∈ V , let Bn = Bn(v0) = v ∈ V | d(v0, v) ≤ n bethe balls, Sn = Sn(v0) = v ∈ V | d(v0, v) = n be the spheres of radius n and σn = |Sn |.There are two versions of the exponential growth defined as

μ(G, v0) := lim supn→∞

1

nlog |Sn | and μ(G, v0) := lim sup

n→∞1

nlog vol(Bn).

Whenever there is a uniform bound on the vertex degree i.e., p = supv∈V |v| < ∞ one easilychecks

μ(G, v0) = μ(G, v0).

In many cases the exponential growth does not depend on the choice of v0. If this is the case,we simply write μ(G) and μ(G).

The growth series for (G, v0) is the formal power series fG,v0(z) = ∑∞n=0 σnzn . By

Cauchy–Hadamard criterion, the growth series represents a well-defined function in theopen complex ball of radius e−μ(G,v0).

Of particular importance in the study of the growth series fG,v0 are recursion formulasfor the sequence σn . In this paper, we consider the case of q-face regular plane tessellationsG = (V, E, F) i.e., all faces are q-gons that is | f | = q for all f ∈ F . Our result will bestated in terms of of normalized average curvatures over spheres

κn := κ(Sn) :=(

2q

q − 2

)1

|Sn |κ(Sn).

Note that the constant 2π(q − 2)/2q is the internal angle of a regular q-gon. Before statingour result we need some more notation: For 3 ≤ q < ∞ let N = q−2

2 if q is even andN = q − 2 if q is odd, and

bl =

⎧⎪⎨

⎪⎩

4q−2 if q is even,

4q−2 if q is odd and l = N−1

2 ,4

q−2 − 2 if q is odd and l = N−12 ,

(4)

for 0 ≤ l ≤ N − 1.

Theorem 2 (Growth recursion formulas) Let G = (V, E, F) be a q-face regular plane tes-sellation without cut locus, Sn = Sn(v0) for some v0 ∈ V and σn = |Sn |. Let κn, N and bl

be defined as above. Then we have the following (N + 1)-step recursion formulas for n ≥ 1:

σn+1 =⎧⎨

⎩

σ1 + ∑n−1l=0 (bl − κn−l)σn−l if n < N ,∑N−1

l=0 (bl − κ N−l)σN−l if n = N ,

−σn−N + ∑N−1l=0 (bl − κn−l)σn−l if n > N .

(5)

123


A proof of this theorem is given in Sect. 3. Note that the constants κk are zero for the regu-lar flat tessellations G3,6, G4,4 and G6,3. The constants κk in (5) can, therefore, be consideredas curvature correction terms for general non-flat tessellations.

In the special case of (p, q)-regular graphs G = G p,q , the terms bl −κk all coincide withthe constant p −2 except in the case if q is odd and l = N−1

2 , when we have bl −κk = p −4.In this case, Theorem 2 is equivalent to the fact that fG,v0 gp,q = h p,q with

h p,q(z) = 1 + 2z + · · · + 2zN + zN+1,

gp,q(z) = 1 − (p − 2)z − · · · − (p − 2)zN + zN+1,

if q is even, and

h p,q(z) = 1 + 2z + · · · + 2zN−1

2 + 4zN+1

2 + 2zN+3

2 + · · · + 2zN + zN+1,

gp,q(z) = 1 − (p − 2)z − · · · − (p − 4)zN+1

2 − · · · − (p − 2)zN + zN+1,

if q is odd. This agrees with results of Cannon and Wagreich [4] and Floyd and Plotnick[10, Sect. 3] that the growth function fG,v0 is the rational function h p,q/gp,q . Moreover, itwas shown in [1,4] that the denominator polynomial gp,q for 1

p + 1q < 1

2 is a reciprocal Salempolynomial, i.e., its roots lie on the complex unit circle except for two positive reciprocal realzeros 1

x p,q< 1 < x p,q < p − 1. This implies that the exponential growth coincides with

log x p,q , i.e.,

μ(G p,q) = μ(G p,q) = log x p,q < log(p − 1) = μ(Tp). (6)

(An even more precise description of the growth of the sequence σn is given in [1, Cor. 3].)Of course, it is desirable to know more about the explicit value of μ(G p,q) = log x p,q . Sincegp,q is divisible by z2 − (p − 4

q−2 )z + 1 in the case q = 3, 4, 6, we have

Proposition 1.1 Let q ∈ 3, 4, 6. Then

μ(G p,q) = μ(G p,q) = log

⎛

⎝ p

2− 2

q − 2+

√(p

2− 2

q − 2

)2

− 1

⎞

⎠ .

In most of the cases the polynomial gp,q is essentially irreducible (expect for some smallwell known factors; see [1, Thm. 1]) and there is no hope to have an explicit expression forits largest zero x p,q > 1. A direct consequence of the isoperimetric inequality in [2, Cor. 5.2]is the following lower estimate of log x p,q :

Proposition 1.2 Let G p,q be non-positively curved, i.e., for all v ∈ V

C = −κ(v) = p

(1

p+ 1

q− 1

2

)≥ 0,

then we have

μ(G p,q) = μ(G p,q) = log x p,q ≥ log

(1 + 2q

q − 1C

). (7)

Note that (7) implies limq→∞ μ(G p,q) = μ(Tp) = log p − 1 for all p ≥ 3.

123


Remark The Mahler measure M(g) of a monic polynomial g ∈ Z[z] with integer coefficientsis given by the product

∏ |zi |, where zi ∈ C are the roots of g of modulus ≥ 1. Lehmer’sconjecture states that for every such g with M(g) > 1 we have

M(g) ≥ M(

1 − z + z3 − z4 + z5 − z6 + z7 − z9 + z10)

≈ 1.1762 . . . .

Thus (7) yields an explicit lower estimate for the Mahler measure of the polynomials gp,q .

Let us now return to general q-face regular tessellations. We conclude from Theorem 2:

Theorem 3 (Curvature/Growth comparison) Let G = (V, E, F) and G = (V , E, F) be twoq-face regular plane tessellations with non-positive vertex curvature, Sn ⊂ V and Sn ⊂ Vbe spheres with respect to the centers v0 ∈ V and v0 ∈ V , respectively, and σn = |Sn | andσn = |Sn |. Assume that the normalized average spherical curvatures satisfy

κ(Sn) ≤ κ(Sn) ≤ 0,

for all n ≥ 0.Then the difference sequence σn −σn ≥ 0 is monotone non-decreasing and, in particular,

we have μ(G) ≥ μ(G). In the case where there is a uniform bound on the vertex degree wealso have μ(G) ≥ μ(G).

A proof of this theorem is given in Sect. 3. (In fact, the proof shows that the vertexcurvature conditions in Theorem 3 can be slightly relaxed: Namely, it suffices that G hasnon-positive vertex curvature and that both graphs G and G are without cut-loci.) This the-orem can be considered as a refined discrete counterpart of the Bishop–Günther–Gromovcomparison theorem for Riemannian manifolds (see, e.g. [13, Theorem 3.101]). The lattercompares volumes of balls in Riemannian manifolds against constant curvature space forms;our discrete counterpart deals with spheres (the result for balls is obtained by adding overspheres) and is more flexible as it allows to use more general comparison spaces.

If we drop the face regularity condition, it is not difficult to derive the following growthcomparison with G replaced by the p-regular tree. The statement holds for arbitrary graphs.

A proof of this result is given in Sect. 3:

Theorem 4 (Tree comparison) Let G be a locally finite, connected graph satisfying |v| ≤ pfor all v ∈ V , for some p ≥ 3. Then

μ(G) = μ(G) ≤ μ(Tp) = log(p − 1),

where Tp is a p-regular tree.

Note that another more involved comparison with a tree was obtained by Higuchi [17]for infinite vertex-regular graphs with each vertex contained a cycle of uniformly boundedlength.

Let us finally discuss some spectral applications (see [27,32] for classical surveys). The(geometric) Laplacian : 2(V, m)−→2(V, m) with m(v) = |v| for v ∈ V is given by

(ϕ

)(v) = 1

|v|∑

w∼v

(ϕ(v) − ϕ(w)),

for all ϕ ∈ 2(V, m), v ∈ V .The relation between the bottom λ0(G) of the spectrum and the bottom λess

0 (G) of theessential spectrum of , the Cheeger constant, and the exponential growth in the discrete

123


case was presented first by Dodziuk and Kendall [8] and Dodziuk and Karp [7]. The bestestimates are due to Fujiwara (see [11,12]):

1 −√

1 − α(G)2 ≤ λ0(G) ≤ λess0 (G) ≤ 1 − 2eμ(G)/2

1 + eμ(G), (8)

Note that the estimates are sharp in the case of regular trees.An immediate consequence of Theorem 1 and (8) is the following combinatorial analogue

of McKean’s theorem (see [25] for the result in the smooth setting):

Corollary 1.3 Let G = (V, E, F) be a locally tessellating planar graph satisfying the ver-tex and face degree bounds in Theorem 1. Moreover, assume that c := infv∈V − 1

|v|κ(v) > 0.Then

1 −√

1 − (2C p,qc

)2 ≤ λ0(G),

where C p,q is defined in (3). This estimate is sharp in the case of regular trees.

Similarly, Theorem 3, (6) and (8) directly imply

Corollary 1.4 Let p, q ≥ 3 and 1p + 1

q ≤ 12 . Let G be a q-face regular tessellation without

cut locus and satisfying |v| ≤ p for all vertices. Then

λess0 (G) ≤ 1 − 2

√x p,q

1 + x p,q.

This estimate is sharp in the case of regular trees.

Let us finish this introduction with some general references. It was shown in [24] thatnon-positive corner curvature implies non-existence of finitely supported eigenfunctions ofall elliptic operators on planar graphs. Lower bounds for the bottom of the essential spectrumin terms of Cheeger constants at infinity or branching rates of general non-planar graphs canbe found in [12,29] for the geometric Laplacian and in [21,33,34] for the physical Laplacian : D() ⊆ 2(V )−→2(V ) given by (ϕ)(v) = ∑

w∼v(ϕ(v)−ϕ(w)), ϕ ∈ D(), v ∈ V .These results show, in particular, the absence of the essential spectrum for graphs with curva-ture converging to negative infinity on complements of finite sets of an increasing exhaustionof G, a phenomenon which was first proved in the context of smooth Riemannian manifoldsby [9]. The different types of Laplacians emerging in this context are due to the choice of themeasure in the corresponding 2 space. For a discussion of the general framework of theseoperators and recent generalizations of the results mentioned above we refer to [22,23].

2 Proof of Theorem 1

Let us first give precise definitions of some notions used in the introduction. Let G = (V, E)

be a planar graph embedded in R2. The faces f of G are the closures of the connectedcomponents in R2\⋃

e∈E e.We further assume that G has no loops, no multiple edges and no vertices of degree one

(leaves). Moreover, we assume that every vertex has finite degree and that every boundedopen set in R2 meets only finitely many faces of G. We call a planar graph with theseproperties simple. We call a sequence of edges e1, . . . , en a walk of length n if there is acorresponding sequence of vertices v1, . . . , vn+1 such that ei = vivi+1. A walk is called a

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path if there is no repetition in the corresponding sequence of vertices v1, . . . , vn . A (finiteor infinite) path with associated vertex sequence . . . vivi+1vi+2 . . . is called a geodesic, if wehave d(vi , v j ) = |i − j | for all pairs of vertices in the path. The boundary of a face f is thesubgraph ∂ f = (V ∩ f, E ∩ f ). We define the degree | f | of a face f ∈ F to be the length ofthe shortest closed walk in the subgraph ∂ f meeting all its vertices. If there is no such finitewalk, we set | f | = ∞. Now we present the conditions that have to be satisfied in order thata planar graph be locally tessellating:

Definition 2.1 A simple planar graph G is called a locally tessellating planar graph if thefollowing conditions are satisfied:

(i) Every edge is contained in precisely two different faces.(ii) Every two faces are either disjoint or have precisely a vertex or a path of edges in

common. In the case that the length of the path is greater than one, then both faces areunbounded.

(iii) Every face is homeomorphic to the closure of an open disc D ⊂ R2, to R2\D or to theupper half plane R × R+ ⊂ R2 and its boundary is a path.

Note that these properties force the graph G to be connected. Examples are tessellationsR2 introduced in [2,3], trees in R2, and particular finite tessellations on the sphere mappedto R2 via stereographic projection.

Now we turn to the proof of Theorem 1. The heart of the proof is Proposition 2.2 below.An earlier version of this proposition in the dual setting is Proposition 2.1 of [2]. We startwith a few preliminary considerations. Let G = (V, E, F) be a locally tessellating planargraph. For a finite set W ⊆ V let GW = (W, EW , FW ) be the subgraph of G induced by W ,where EW are the edges in E with both end points in W and FW are the faces induced bythe graph (W, EW ). Euler’s formula states for a finite and connected subgraph GW (observethat FW contains also the unbounded face):

|W | − |EW | + |FW | = 2. (9)

Recall that ∂E W is the set of edges connecting a vertex in W with one in V \W . By ∂F W , wedenote the set of faces in F which contain an edge of ∂E W . Moreover, we define the innerdegree of a face f ∈ ∂F W by

| f |iW = | f ∩ W |.

We will need two useful formulas which hold for arbitrary finite and connected subgraphsGW = (W, EW , FW ). The first formula is easy to see and reads as

∑

v∈W

|v| = 2|EW | + |∂E W |. (10)

Since W is finite, the set FW contains at least one face which is not in F , namely the unboundedface surrounding GW , but there can be more. Define c(W ) as the number

c(W ) = |FW | − |FW ∩ F | ≥ 1. (11)

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Note that |FW ∩ F | is the number of faces in F which are entirely enclosed by edges of EW .Sorting the following sum over vertices according to faces gives the second formula

∑

v∈W

∑

f v

1

| f | = |FW ∩ F | +∑

f ∈∂F W

| f |iW| f |

= |FW | − c(W ) +∑

f ∈∂F W

| f |iW| f | . (12)

Proposition 2.2 Let G = (V, E, F) be a locally tessellating planar graph and W ⊂ V bea finite set of vertices such that the induced subgraph GW is connected. Then we have

κ(W ) = 2 − c(W ) − |∂E W |2

+∑

f ∈∂F W

| f |iW| f | .

Proof By the Eqs. (9), (10) and (12) we conclude

κ(W ) =∑

v∈W

⎛

⎝1 − |v|2

+∑

f v

1

| f |

⎞

⎠

= |W | − |EW | − |∂E W |2

+ |FW | − c(W ) +∑

f ∈∂F W

| f |iW| f |

= 2 − c(W ) − |∂E W |2

+∑

f ∈∂F W

| f |iW| f | .

A finite set W ⊂ V is called a polygon, if GW is connected and if c(W ) = 1. This notion

becomes understandable if one looks at the dual setting: Every vertex v ∈ W corresponds toa face f ∗(v) ∈ F∗ in the dual planar graph G∗ = (V ∗, E∗, F∗), and W ⊂ V is a polygon ifand only if

⋃v∈W f ∗(v) ⊂ R2 is homeomorphic to a closed disc (here f denotes the closure

of the geometric realization of the face f ). For v ∈ W , let |v|eW denote the number of edgesin ∂E W adjacent to v and call it the external degree of v (w.r.t. W ). Moreover, let ∂V W bethe set of vertices in W with |v|eW ≥ 1.

Proposition 2.3 Ler G = (V, E, F) be a locally tessellating planar graph satisfying thevertex and face degree bounds in Theorem 1 and W ⊂ V be a polygon with |v|eW ≤ p − 2for all v ∈ ∂V W . Then we have

|∂E W | ≥ 2C p,q(1 − κ(W )). (13)

Moreover, under the assumption of (a) or (b) in Theorem 1, we have

|∂E W ||W | ≥ 2C p,qC or

|∂E W |vol(W )

≥ 2C p,qc, respectively.

Proof Observe first that we have the inequality∑

f ∈∂F W

| f |iW ≥ |∂V W | + |∂E W |. (14)

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This can be seen as follows: Every face f ∈ ∂F W may have some edges and some isolatedvertices in common with the induced graph GW = (W, EW , FW ). Since the vertices of ∂V Ware connected in GW , there are at least |∂V W | pairs ( f, e) ∈ ∂F W × E with e ∈ ∂ f ∩ EW .These pairs contribute at least 2|∂V W | to the left hand sum in (14). At every vertex v ∈ ∂V W ,there are |v|eW − 1 faces of ∂F W which meet GW in the isolated vertex v. Adding over allthese vertices v ∈ ∂V W , we obtain the total contribution |∂E W | − |∂V W | to the left handsum in (14). One easily checks that there is no overlap of both contributions, leading to theabove inequality.

Using (14), |∂V W | ≥ 1p−2 |∂E W |, | f | ≤ q for all f ∈ F , and Proposition 2.2, we obtain

|∂E W |(

1

2− p − 1

q(p − 2)

)≥ |∂E W |

2−

∑

f ∈∂F W

| f |iW| f | = 1 − κ(W ),

which yields (13). The second formula of the proposition follows from −κ(W ) ≥ C |W | incase (a) and from −κ(W ) ≥ c vol(W ) in case (b).

Henceforth, let G = (V, E, F) be a locally tessellating planar graph as in Theorem 1.Recall that C p,q = q(p−2)

(p−2)(q−2)−2 . The conditions |v| ≤ p and | f | ≤ q for all v ∈ V and

f ∈ F imply 2C p,qC ≤ p − 2 and 2C p,qc ≤ p−2p .

Lemma 2.4 Let v ∈ V and W = v. Then we have

|∂E W ||W | = |v| ≥ 2C p,qC (15)

and

|∂E W |vol(W )

= 1 ≥ p − 2

p.

Proof The only non trivial inequality is (15). It follows in a straightforward way fromκ(v) ≤ −C that

C ≤ q − 2

2q|v| − 1.

This implies that

2C p,qC ≤ (p − 2)(q − 2)

(p − 2)(q − 2) − 2|v| − 2q(p − 2)

(p − 2)(q − 2) − 2

≤ |v| − 2

(p − 2)(q − 2) − 2(q(p − 2) − p).

The lemma follows now from the fact that q(p − 2) − p ≥ 0 for p, q ≥ 3. Lemma 2.5 Assume that there is a finite set W ⊂ V such that

|∂E W ||W | < 2C p,qC ≤ p − 2 or

|∂E W |vol(W )

< 2C p,qc ≤ p − 2

p, respectively. (16)

Then there exists a polygon W ′ ⊂ V with |v|eW ′ ≤ p − 2 for all v ∈ ∂V W ′, such that

|∂E W ′||W ′| ≤ |∂E W |

|W | or|∂E W ′|vol(W ′)

≤ |∂E W |vol(W )

, respectively.

123


Proof Observe first that we can always find a non-empty subset W0 ⊆ W such that GW0 isa connected component of GW and that |∂E W0|/|W0| ≤ |∂E W |/|W | or |∂E W0|/vol(W0) ≤|∂E W |/vol(W ), respectively. Note that GW0 has only one unbounded face. By adding allvertices of V contained in the union of all bounded faces of GW0 , we obtain a polygon witheven smaller isoperimetric constants. Let us denote this non-empty polygon, again, by W .By Lemma 2.4, W must have at least two vertices. By connectedness of GW and by theinequality |W | ≥ 2, we have |v|eW ≤ p −1 for all v ∈ W . Assume there is a vertex v ∈ ∂V Wwith |v|eW = p−1. Let W ′ := W\v. Then one easily checks that the condition (16) implies

|∂E W ′||W ′| = |∂E W | + 2 − p

|W | − 1<

|∂E W ||W |

or

|∂E W ′|vol(W ′)

= |∂E W | + 2 − p

vol(W ) − p<

|∂E W |vol(W )

,

respectively. Repeating this elimination of vertices with external degree p − 1, we end upwith a polygon W ′ satisfying |v|eW ′ ≤ p − 2 for all v ∈ W ′ or with W ′ equal to a singlevertex. But the latter cannot happen by Lemma 2.4. Proof of Theorem 1 Since 2C p,qC ≤ p − 2 or 2C p,qc ≤ p−2

p , we only have to consider

the cases when α(G) < p − 2 or α(G) <p−2

p , since otherwise there is nothing to prove.Lemma 2.5 states that, if there is a finite W ⊂ V with

|∂E W ||W | < 2C p,qC or

|∂E W |vol(W )

< 2C p,qc, respectively,

then there is a polygon W ′ with |v|eW ′ ≤ p−2 for all v ∈ ∂V W ′ satisfying the same inequality.But this contradicts Proposition 2.3, finishing the proof of Theorem 1.

3 Proofs of Theorems 2, 3 and 4

Let G = (V, E, F) be a q-face regular plane tessellation without cut locus, v0 ∈ V, Sn =Sn(v0) and σn = |Sn |. Recall that we have N = q−2

2 if q is even and N = q − 2 if q is odd.The recursion formulas (5) in Theorem 2 for n ≤ N and n > N , respectively, require separateproofs. However, both proofs are based on the following results from [2, Sect. 6]. (Note thatthese results are presented there in the dual setting of vertex-regular graphs.) Proposition 6.3in [2] states for n ≥ 1 that

κ(Bn) = 1 − q − 2

2q(σn+1 − σn) +

q−2∑

j=2

q − 2 j

2qc j

n , (17)

where Bn = v ∈ V | d(v0, v) ≤ n denotes the ball and

c jn = | f ∈ F | | f ∩ (V \Bn)| = j|

for 1 ≤ j ≤ q − 1. Moreover from Lemma 6.2 in [2] we have the following recurrencerelations for c j

n , n ≥ 1, which arise very naturally from the geometric context

(i) cln = cl+2

n−1, for 1 ≤ l ≤ q − 3,

(ii) cq−2n = c2

n−1,

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(iii) cq−1n = c1

n + σn+1 − σn = c3n−1 + σn+1 − σn .

We first proof of (5) for n ≤ N . Let τ : 0, 1, . . . , N → 1, 2, . . . , q − 1 be defined as

τ(k) =⎧⎨

⎩

q − 1 − 2k if q is even and 0 ≤ k ≤ N ,

q − 1 − 2k if q is odd and 0 ≤ k ≤ N−12 ,

2q − 3 − 2k if q is odd and N+12 ≤ k ≤ N .

Note that τ is defined precisely in such a way that we have

cτ(k+1)n+1 = cτ(k)

n for 0 ≤ k ≤ N − 1 and n ≥ 0, (18)

by the recurrence relations (i) and (ii).

Lemma 3.1 Let 1 ≤ n ≤ N. Then we have

cτ(l)n =

⎧⎨

⎩

σn+1−l − σn−l for 1 ≤ l ≤ n − 1,

σ1 for l = n,

0 for n + 1 ≤ l ≤ N .

Moreover, in the case of even q, we have c2ln = 0 for 1 ≤ l ≤ N.

Proof One easily sees that c j0 = 0 for 1 ≤ j ≤ q − 2 and cq−1

0 = σ1. The recurrencerelations (i) and (ii) imply that c1

k = 0 for 0 ≤ k ≤ N − 1 and c1N = σ1. Using (iii), we

obtain cq−1k = σk+1 − σk for 1 ≤ k ≤ N − 1. The value of cτ(l)

n can now be deduced fromthese results by repeatedly applying (18) in each of the cases 1 ≤ l ≤ n − 1, l = n andn + 1 ≤ l ≤ N . Lemma 3.2 Let N ≥ 2, 1 ≤ n ≤ N and bl be defined as in (4). Then we have

q − 6

q − 2σn +

q−2∑

j=2

q − 2 j

q − 2c j

n =⎧⎨

⎩

∑n−1l=1 blσn−l if n ≤ N − 1,

6−qq−2σ1 + ∑N−2

l=1 blσN−l if n = N .

Proof First observe that, since 2 ≤ τ(l) ≤ q − 2 for 1 ≤ l ≤ N − 1 and cτ(l)n = 0 for

n + 1 ≤ l ≤ N by Lemma 3.1

q−2∑

j=2

(q − 2 j)c jn =

∑nl=1(q − 2τ(l))cτ(l)

n if n ≤ N − 1,∑N−1l=1 (q − 2τ(l))cτ(l)

n if n = N ,

(Note for the case n = N : we have 2 ≤ τ(l) ≤ q − 2 only for 1 ≤ l ≤ N − 1 and τ(N ) = 1.This makes it necessary to treat this case separately.) The proof is now straightforward usingthe help of Lemma 3.1 and the equation (q − 2)bl = 2(τ (l) − τ(l + 1)). Proof of Theorem 2 We rewrite Eq. (17) as follows:

σn+1 − σn = 2q

q − 2− 2q

q − 2

n∑

l=0

κ(Sl) +q−2∑

j=2

q − 2 j

q − 2c j

n .

Using κ(S0) = 1 − q−22q σ1 and 2q

q−2κ(Sl) = κlσl we obtain

σn+1 =(

σ1 −n∑

l=1

κlσl

)+ σn +

q−2∑

j=2

q − 2 j

q − 2c j

n . (19)

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The recursion formulas in Theorem 2 for N ≥ 2 and n ≤ N follow now directly from(19) and Lemma 3.2. The case n = N = 1 has to be treated separately: In this case we have∑q−2

j=2(q − 2 j)c jn = 0 and (19) simplifies to σ2 = (2 − κ1)σ1. The result follows now from

the fact that b0 = 2.It remains to prove the recursion formula (5) for n > N . We first consider the case N ≥ 2.

Repeated application of the recurrence relations (i)–(iii) yields

q−2∑

j=2

q − 2 j

q − 2c j

n =(

N−1∑

l=0

blσn−l

)− (σn + σn−(N−1)) +

q−2∑

j=2

q − 2 j

q − 2c j

n−N . (20)

Since 2qq−2 (κ(Bn) − κ(Bn−N )) = ∑N−1

l=0 κn−lσn−l , we obtain using (17) and (20)

N−1∑

l=0

κn−lσn−l = − (σn+1 − σn) + (σn−(N−1) − σn−N

)

+(

N−1∑

l=0

blσn−l

)− (

σn + σn−(N−1)

),

which immediately yields the recursion formula (5) for n > N ≥ 2. The case N = 1 isparticularly easy and is left to the reader. This finishes the proof of Theorem 2.

Now we turn to the proof of Theorem 3. Note first that non-positive vertex curvatureimplies non-positive corner curvature in the case of face-regular graphs. By [3, Theorem 1],both graphs G, G are without cut-loci and we can apply the recursion formulas in Theorem 2.

Lemma 3.3 We have the following estimates for 0 ≤ l ≤ N − 1 and k ≥ 1:

bl − κk ≥ 2, if l = N − 1

2(= 0) and q ∈ 3, 4,

bl − κk ≥ 1, if l = N − 1

2or q even,

bl − κk ≥ 0, if l = N − 1

2(= 1) and q = 5,

bl − κk ≥ −1, if l = N − 1

2and q ≥ 7 odd.

Proof The case “l = N−12 or q even” follows from bl = 4

q−2 and κ(v) = 1 − q−22q |v| ≤

1 − 3 q−22q .

Now assume that l = N−12 . Since bl ≥ 4

q−2 − 2, the previous considerations lead tobl − κk ≥ −1. If q = 3 or q = 4, then bl = 2 and, consequently, bl − κk ≥ bl = 2. Finally,if q = 5, then κ(v) ≤ 0 implies that |v| ≥ 4 and thus κ(v) ≤ 1 − 4 q−2

2q . Using this fact leadsdirectly to bl − κk ≥ 0.

From the above lemma we deduce the following inequalities:

Lemma 3.4 We have

(a) b0 − κk ≥ 1.(b) bN−1 − κk ≥ 1 if N ≥ 2.(c) b0 − κk ≥ 2 if q = 3 or q = 4.

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(d) Let n, N ≥ 1, 1 ≤ k ≤ minn, N , and assume that γi ≥ 0 are mononote non-decreas-ing for n − k + 1 ≤ i ≤ n − 1. Then

k−1∑

l=1

(bl − κn−l)γn−l ≥ 0.

Proof (a), (b) and (c) are trivial consequences of Lemma 3.3. (d) follows immediately fromLemma 3.3, unless we have k − 1 ≥ N−1

2 and q ≥ 7 odd. But in this case we have N−12 ≥ 2

and

k−1∑

l=1

(bl − κn−l)γn−l ≥ (b1 − κn−1)γn−1 + (b N−12

− κn− N−12

)γn− N−12

≥ γn−1 − γn− N−12

≥ 0,

by the monotonicity of γi . Proof of Theorem 3 Let κn = κ(Sn). Then the condition κ(Sn) ≤ κ(Sn) ≤ 0 reads κn ≤ κn

for n ≥ 0. Note that σ0 = σ0 = 1. Since σ1 = 2qq−2 − κ0, σ1 = 2q

q−2 − κ0 we conclude that

σ1 − σ1 = κ0 − κ0 ≥ 0 = σ0 − σ0.

The proof is based on induction over n: Assume that n ≥ 1 and that γk := σk − σk isnon-negative and monotone non-decreasing for 0 ≤ k ≤ n. We aim to show that γn+1 ≥ γn .

We first consider the case n ≤ N . Then the recursion formula (5) yields

γn+1 ≥ γn +n−1∑

l=1

(bl − κn−l)γn−l , (21)

and γn+1 ≥ γn follows from Lemma 3.4(d).Finally, we consider the case n > N . If N ≥ 2, the recursion formula (5), Lemma 3.4(a,b)

and the monotonicity of γk yields

γn+1 ≥ γn +(

N−1∑

l=1

(bl − κn−l)γn−l

)− γn−N

≥ γn +N−2∑

l=1

(bl − κn−l)γn−l .

Again, γn+1 ≥ γn follows now from Lemma 3.4(d). If N = 1 (i.e., q = 3 or q = 4), therecursion formula (5) simplifies considerably and, using Lemma 3.4(c), we conclude that

γn+1 ≥ 2γn − γn−1 ≥ γn,

finishing the proof of Theorem 3. Proof of Theorem 4 Let v0 ∈ V be arbitrary and Sn, n ∈ N0 the corresponding distancespheres. Since we assumed G to be connected and the vertex degree to be uniformly boundedfrom above we have μ(G) = μ(G, v0) = μ(G, v0) = μ(G). Let T be a spanning tree of Gwhich leaves the distance relation with respect to v0 invariant. This can easily be achieved asfollows: One removes all edges which connect vertices within a sphere. Further one removesinductively all edges which connect a vertex in Sn to vertices in Sn−1 except for one edge.

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Since the sphere structure is left invariant the spanning tree T has the same exponential vol-ume growth as G, i.e., μ(G, v0) = μ(T, v0). Moreover T can be embedded into a p-regulartree Tp since the vertex degree in T is smaller or equal than the supremum of the vertexdegrees in G which we denoted by p. This finishes the proof. Acknowledgments Matthias Keller likes to thank Daniel Lenz who encouraged him to study the connec-tion between curvature and spectral theory. Matthias Keller was financially supported during this work bySDW. Norbert Peyerimhoff is grateful for the financial support of the Technical University of Chemnitz. Bothauthors like to thank Ruth Kellerhals and Victor Abrashkin for very useful discussions. Moreover the authorsvery much appreciate the comments of Radosław Wojciechowski which let to a simplification of the proof ofTheorem 4.

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27. Mohar, B., Woess, W.: A survey on spectra of infinite graphs. Bull. Lond. Math. Soc. 21(3), 209–234 (1989)28. Stone, D.A.: A combinatorial analogue of a theorem of Myers and Correction to my paper: “A combina-

torial analogue of a theorem of Myers”. Illinois J. Math. 20(1), 12–21 (1976); 20(3), 551–554 (1976)29. Urakawa, H.: The spectrum of an infinite graph. Can. J. Math. 52(5), 1057–1084 (2000)30. Weber, A.: Analysis of the physical Laplacian and the heat flow on a locally finite graph. http://arxiv.org/

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CHAPTER 15

M. Keller, Curvature, geometry and spectralproperties of planar graphs, Discrete &

Computational Geometry 46 (2011), 500–525.

329

Discrete Comput Geom (2011) 46:500–525DOI 10.1007/s00454-011-9333-0

Curvature, Geometry and Spectral Properties of PlanarGraphs

Matthias Keller

Received: 4 May 2010 / Revised: 15 January 2011 / Accepted: 9 February 2011 /Published online: 1 March 2011© Springer Science+Business Media, LLC 2011

Abstract We introduce a curvature function for planar graphs to study the connec-tion between the curvature and the geometric and spectral properties of the graph. Weshow that non-positive curvature implies that the graph is infinite and locally similarto a tessellation. We use this to extend several results known for tessellations to gen-eral planar graphs. For non-positive curvature, we show that the graph admits no cutlocus and we give a description of the boundary structure of distance balls. For neg-ative curvature, we prove that the interiors of minimal bigons are empty and deriveexplicit bounds for the growth of distance balls and Cheeger’s constant. The latter areused to obtain lower bounds for the bottom of the spectrum of the discrete Laplaceoperator. Moreover, we give a characterization for triviality of the essential spectrumby uniform decrease of the curvature. Finally, we show that non-positive curvatureimplies the absence of finitely supported eigenfunctions for nearest neighbor opera-tors.

Keywords Discrete curvature · Hyperbolic properties · Cut locus · Cheeger’sconstant · Exponential growth · Spectrum of graphs · Unique continuation ofeigenfunctions

1 Introduction

There is a long tradition of studying planar graphs, i.e., graphs that can be embeddedin a topological surface homeomorphic to R2 without self intersection. In particular,the curvature of tessellating graphs received a great deal of attention during the recentyears, see for instance [2–4, 9, 11, 15, 19, 20, 25–27, 30] and references therein. Here,we introduce a notion of curvature for general planar graphs to study their geometry

M. Keller ()Mathematical Institute, Friedrich-Schiller-University Jena, 07743 Jena, Germanye-mail: [email protected]

Discrete Comput Geom (2011) 46:500–525 501

and spectral properties. In the case of tessellations, which are also called tilings, thedefinitions coincide with the one of [2, 3, 15, 25, 27].

We next summarize and discuss the main results of the paper. Precise formulationscan be found in the sections, where the results are proven.

The key insight of this paper is that non-positive curvature alone has already verystrong implications on the structure of a planar graph, i.e.:

(1) The graph is infinite and locally tessellating (Theorem 1 in Sect. 3).(2) The graph is locally similar to a tessellation (Theorem 2 in Sect. 4).

Let us comment on these points. In contrast to the statement about the infinity, posi-tive curvature implies finiteness of the graph. This was studied in [4, 9, 15, 25, 26].Note also that locally tessellating graphs, recently introduced in [20] (see also [30]),allow for a unified treatment of planar tessellations and trees. The idea of (2) is toconstruct an embedding of a locally tessellating graph into a tessellation that pre-serves crucial properties of a fixed finite subset. This embedding allows us to carryover many results known for tessellations to planar graphs. Indeed, most of the resultsare then direct corollaries of (1), (2) and the corresponding results for tessellations.In particular, we obtain for planar graphs with non-positive curvature:

(3) Absence of cut locus, i.e., every distance minimizing path can be continued toinfinity (Theorem 3 in Sect. 5.1).

(4) A description of the boundary of distance balls (Theorem 4 in Sect. 5.2).

For regular tessellations a result similar to (3) was obtained by Baues, Peyerimhoffin [2] and was later extended to general tessellations by the same authors in [3]. Thisresult is considered a discrete analog of the Hadamard–Cartan theorem in differentialgeometry. Questions about the possible depth of the cut locus for Cayley graphs werestudied under the name dead-end-depth in [5], see also references therein. The bound-ary structure of distance balls, mentioned in (4), plays a crucial role in the techniquesof Zuk [30]. These concepts were later refined by [2, 3] under the name admissibil-ity of distance balls. Here, we generalize the most important statements of [3, 30] toplanar graphs.

For negatively curved planar graphs, we prove the following:

(5) Bounds for the growth of distance balls (Theorem 5 in Sect. 5.3).(6) Positivity and bounds for Cheeger’s constant (Theorem 6 in Sect. 5.4).(7) Empty interior for minimal bigons (Theorem 7 in Sect. 5.5).

In [2], a result similar to (5) is shown for tessellations. As for (6), positivity ofCheeger’s constant was proven by Woess [27] for tessellations and simultaneouslyby Zuk in a slightly more general context. Later, this was rediscovered in [15]. Moreprecisely, Zuk showed that the norm of the transition matrix is strictly smaller thanone. However, this is equivalent to positivity of Cheeger’s constant, see [7, 8, 14].Here, we give new explicit bounds for Cheeger’s constant, which can be interpretedin terms of the curvature. For related results, see for instance [16, 17, 19, 20]. Theresults (6) and (7) can be understood as hyperbolic properties of a graph. In particu-lar, if G is the Cayley graph of a finitely generated group both results imply Gromovhyperbolicity. In general, this is not the case in our situation. However, under the ad-ditional assumption that the number of edges of finite polygons is uniformly bounded,

502 Discrete Comput Geom (2011) 46:500–525

empty interior for minimal bigons, along with Gromov hyperbolicity can be proven,see [3, 30].

Let us now turn to the spectral implications. It is well known that bounds forCheeger’s constant imply bounds for the bottom of the spectrum of the graph Lapla-cian, see [1, 6, 8, 12, 22]. The Cheeger constant also plays an important role in ran-dom walks. In particular, the simple random walk is transient if Cheeger’s constantis positive, see [14, 27, 28] and also [24]. Along with these well known relations, weshow for planar graphs with non-positive curvature:

(8) Triviality of the essential spectrum of the Laplacian if and only if the curvaturedecreases uniformly to −∞ (Theorem 8 in Sect. 6.1).

(9) Absence of finitely supported eigenfunctions for nearest neighbor operators(Theorem 9 in Sect. 6.2).

Statement (8) generalizes results of [19] for tessellations and of [12] for trees. State-ment (9) is an extension of [21]. As for general planar graphs eigenfunctions of finitesupport can occur, this unique continuation statement shows that the analogy betweenRiemannian manifolds and graphs is much stronger in the case of non-positive cur-vature.

The paper is structured as follows. In Sect. 2, we give the basic definitions. Sec-tion 3 is devoted to the statement and the proof of Theorem 1. In Sect. 4, we constructthe embedding into tessellations. Sect. 5 concentrates on the geometric applications(3)–(7) and Sect. 6 is devoted to spectral applications (8) and (9).

2 Definitions and a Combinatorial Gauss–Bonnet Formula

Let G = (V ,E) be a graph embedded in an oriented topological surface S . An em-bedding is a continuous one-to-one mapping from a topological realization of G

into S . A graph admitting such an embedding into R2 is referred to as planar. Wewill identify G with its image in S . The faces F of G are defined as the closures ofthe connected components of S \ ⋃

E. We write G = (V ,E,F ).We call G simple if it has no loops (i.e., no edge contains only one vertex) and no

multiple edges (i.e., two vertices are not connected by more than one edge). We saythat G is locally finite if for every point in S there exists an open neighborhood ofthis point that intersects with only finitely many edges. A characterization whethera Cayley graph of a group allows for a planar embedding that is locally finite wasrecently given in [13].

For the rest of this paper, we will assume that G is locally finite.We call two distinct vertices v,w ∈ V adjacent and we write v ∼ w if there is

an edge containing both of them. We also say that two elements of V , E and F

(possibly of distinct type) are adjacent if their intersection is non-empty and we calltwo adjacent elements of the same type neighbors.

By the embedding, each edge corresponds uniquely to a curve (up to parametriza-tion). We call a sequence of edges a walk if the corresponding subsequent curves canbe composed to a curve. Here, we allow for two sided infinite sequences. The lengthof the walk is the number of elements of the sequence, whenever there are only fi-nitely many and infinite otherwise. We refer to a subgraph, where the edges can be


ordered to form a walk and each vertex is contained in exactly two edges as a path.The length of a path is the length of the shortest walk passing all vertices. We callvertices that are contained in two edges of a walk or a path the inner vertices of thewalk or the path. A walk or a path is called closed if every vertex contained in it is aninner vertex and simply closed if all vertices are contained in exactly two edges. Notethat every simply closed walk induces a path and every closed path is simply closed.We say a graph is connected if any two vertices can be joined by a path.

For the remainder of this paper, we will assume that G is connected.For a face f ∈ F , we call a walk a boundary walk of f if it meets all vertices in

f and if it is closed whenever it is finite. The existence of boundary walks for allfaces is ensured by the connectedness of the graph. We let the degree |f | of a facef ∈ F be the length of the shortest boundary walk of f , whenever f admits a finiteboundary walk and infinite otherwise. We define the degree |e| of an edge e ∈ E asthe number of vertices contained in e. For a vertex v ∈ V , we define the degree by

|v| := 2∑

e∈E,v∈e

1

|e| .

The formula can be interpreted as the number of adjacent edges with degree two plustwice the number of adjacent edges with degree one. The latter are counted twicesince they meet v twice. A vertex v ∈ V with |v| = 1 is called a terminal vertex.

To define curvature functions, we first have to introduce the corners of a planargraph G. The set of corners C(G) is a subset of V × F such that the elements (v, f )

satisfy v ∈ f . We denote for a vertex v the set Cv(G) := (v, g) ∈ C(G) and fora face f the set Cf (G) := (w,f ) ∈ C(G). The degree or the multiplicity |(v, f )|of a corner (v, f ) ∈ C(G) is the minimal number of times the vertex v is met by aboundary path of f . For tessellations, the definition of corners coincides with the onein [2, 3] and there every corner has degree one. However, if G is not a tessellation thedegree of a corner can be larger than one (see for instance the left hand side of Fig. 1in the next section). For a vertex v ∈ V and a face f ∈ F , we have

|v| =∑

(v,g)∈Cv(G)

∣∣(v, g)

∣∣ and |f | =

∑

(w,f )∈Cf (G)

∣∣(w,f )

∣∣.

We can think of the corners of a vertex v with respect to the multiplicity as the parti-tions of a sufficiently small ball after removing the edges adjacent to v (where smallmeans that no edge is completely included in the ball and there are exactly |v| parti-tions).

The curvature function on the corners κGC : C(G) → R is defined by

κGC (v,f ) := 1

|v| − 1

2+ 1

|f | ,

with the convention that 1/|f | = 0 whenever |f | = ∞. We define the curvature func-tion on the vertices κG

V : V → R by

κGV (v) :=

∑

(v,f )∈Cv(G)

∣∣(v, f )

∣∣κG

C (v,f ).


By direct calculation, we arrive at

κGV (v) = 1 − |v|

2+

∑

(v,f )∈Cv(G)

∣∣(v, f )

∣∣ 1

|f | .

We define the curvature function on the faces κGF : F → [−∞,∞) by

κGF (f ) :=

∑

(v,f )∈Cf (G)

∣∣(v, f )

∣∣κG

C (v,f ).

If |f | = ∞ and if there are infinitely many vertices in f with vertex degree of at leastthree, then the curvature of a face f takes the value −∞.

For finite subsets V ′ ⊆ V , F ′ ⊆ F , we write

κGV (V ′) :=

∑

v∈V ′κGV (v) and κG

F (F ′) :=∑

f ∈F ′κGF (f ).

Moreover, we let

κC(G) := sup(v,f )∈C(G)

κGC (v,f ), κV (G) := sup

v∈V

κGV (v), κF (G) := sup

f ∈F

κGF (f ).

We call a face a polygon if it is homeomorphic to the closure of the unit disc D in R2

and its boundary is a closed path. We call a face an infinigon if it is homeomorphic toR2 \ D or the upper half-plane R × R+ ⊂ R2 and its boundary is a path. A graph G

is called tessellating if the following conditions are satisfied:

(T1) Every edge is contained in precisely two different faces.(T2) Every two faces are either disjoint or intersect precisely in a vertex or in an

edge.(T3) Every face is a polygon.

The additional assumption in [2, 3] that each vertex has finite degree is already im-plied by local finiteness of G. Note that the dual graph of a non-positively cornercurved tessellation is also a non-positively corner curved tessellation. For a discus-sion, we refer to [2].

We call a path of at least two edges an extended edge if all inner vertices havedegree two and the beginning and ending vertex, in the case that they exist, havedegree greater than two. An extended edge contained in two faces with infinite degreeis called regular. We introduce two weaker conditions than (T2) and (T3):

(T2∗) Every two faces are either disjoint or intersect precisely in a vertex, an edge ora regular extended edge.

(T3∗) Every face is a polygon or an infinigon.

We call a graph satisfying (T1), (T2), (T3∗) strictly locally tessellating and a graphsatisfying (T1), (T2∗), (T3∗) locally tessellating. Note that a locally tessellating graphthat contains no extended edge is strictly locally tessellating. While the graphs studiedin [2–4, 9, 15, 17, 19, 21, 24–27] are tessellating, the results of [20, 30] concernstrictly locally tessellating graphs. We give some examples.


Examples (1) Tessellations of the plane R2 or the sphere S2 are strictly locally tes-sellating. Furthermore, tessellations of S2 embedded into R2 via stereographic pro-jection are also strictly locally tessellating.

(2) Trees are strictly locally tessellating if and only if the branching number ofevery vertex is greater than one. In this case, the tree is negatively curved in eachcorner, vertex and face.

(3) The graph with vertex set Z and edge set [n,n+ 1]n∈Z is locally tessellating,but not strictly locally tessellating and has curvature zero in every corner, vertex andface.

Next, we will prove a combinatorial Gauss–Bonnet formula. We refer to [2] forbackground and proof in the case of tessellations, see also [4, 9] for further reference.

Let G = (V ,E,F ) be a planar graph embedded into S2 or R2. For a subsetW ⊆ V , we denote by GW = (W,EW ,FW) the subgraph of G induced by the vertexset W , where EW ⊆ E are the edges that contain only vertices in W and FW arethe faces induced by the graph (W,EW). For a finite connected subset of verticesW ⊆ V , Euler’s formula reads

|VW | − |EW | + |FW | = 2.

Observe that FW contains also an unbounded face, which explains the two on theright hand side.

Proposition 1 (Gauss–Bonnet formula) Let G be planar and W ⊆ V finite and con-nected. Then

κGW

V (W) = 2.

Proof We have, by definition,

κGW

V (W) =∑

(v,f )∈C(GW )

∣∣(v, f )

∣∣(

1

|v| + 1

|f | − 1

2

)

=∑

v∈W

∑

c∈Cv(GW )

|c||v| +

∑

f ∈FW

∑

c∈Cf (GW )

|c||f | −

∑

v∈W

∑

c∈Cv(GW )

|c|2

.

Since |v| = ∑c∈Cv(GW ) |c|, the first term is equal to |W |. As W is connected, we have

|f | = ∑c∈Cf (GW ) |c|. Thus, the second term is equal to |FW |. Let Ev,j = e ∈ EW |

v ∈ e, |e| = j for v ∈ W and j = 1,2. Then,∑

c∈Cv(GW ) |c| = |v| = 2|Ev,1| + |Ev,2|by the definition of |v|. Moreover, for each e ∈ Ev,2 there is a unique w ∈ W , w = v,such that e ∈ Ew,2. We conclude

∑

v∈W

∑

c∈Cv(GW )

|c|2

=∑

v∈W

|Ev,1| + 1

2|Ev,2| = |EW |.

By Euler’s formula, we obtain the result.


Remark The Gauss–Bonnet formula immediately implies that a finite graph mustadmit some positive curvature. This is, in particular, the case for locally finite planargraphs embedded in S2.

We write d(v,w) for the length of the shortest path connecting the vertices v

and w. For a set of vertices W ⊆ V , we define the balls and the spheres of radius n

by

Bn(W) := BGn (W) :=

v ∈ V | d(v,w) ≤ n for some w ∈ W,

Sn(W) := SGn (W) := Bn(W) \ Bn−1(W).

We define the boundary faces of W by

∂F W := ∂GF W :=

f ∈ F | f ∩ W = ∅, f ∩ V \ W = ∅.

3 Infinity and Local Tessellating Properties of Non-positively Curved PlanarGraphs

This section is dedicated to prove the first main result.

Theorem 1 Let G be a planar graph that is connected and locally finite. If one ofthe following conditions is satisfied

(a) κC(G) ≤ 0 or(b) κV (G) ≤ 0 and G is simple or(c) κF (G) ≤ 0, each extended edge is regular and there are no terminal vertices,

then G is infinite and locally tessellating. If a strict inequality holds in condition (a)or (b) then G is even strictly locally tessellating.

The main idea of the proof can be summarized as follows. The infinity is a directconsequence of the Gauss–Bonnet formula. Moreover, assuming a graph is not locallytessellating, we will construct a finite planar graph with smaller or equal curvature,which can be embedded into S2. By the Gauss–Bonnet formula, this graph must admitsome positive curvature. As the curvature is smaller or equal, the original graph musthave some positive curvature as well.

We start with analyzing the pathologies that occur in general planar graphs thatare not locally tessellating. Let G = (V ,F,E) be a planar graph that is connectedand locally finite. A face f ∈ F is called degenerate if it contains a vertex v suchthat |(v, f )| ≥ 2 for the corner (v, f ). Note that this is in particular the case if v iscontained in three or more boundary edges of f or there is an edge that is included inno other face except for f . A pair of faces (f, g) is called degenerate if f ∩g consistsof at least two connected components. Figure 1 shows examples of degenerate faces.We denote by D(F) ⊆ F the set of all faces that are degenerate or are contained in adegenerate pair of faces.


Fig. 1 Examples of a degenerate face f and a degenerate pair of faces (f, g)

Fig. 2 The figure shows how to isolate a finite subset W , whenever there is degenerate pair of faces or adegenerate face

Lemma 1 Let G = (V ,E,F ) be a simple, planar graph that is connected, locallyfinite and contains no terminal vertices. If D(F) = ∅, then there is a finite and con-nected subgraph G1 = (V1,E1), which is bounded by a simply closed path and sat-isfies the following property: there are at most two vertices v1, v2 ∈ V1 such that eachpath connecting V1 and V \ V1 meets either v1 or v2. In particular, v1, v2 lie in theboundary path of G1.

Proof The proof consists of two steps. Firstly, we find a finite set W ⊆ V that con-tains at most two vertices with the asserted property. We have to deal with the case ofa degenerate face and a degenerate pair separately. Secondly, we find a subgraph ofGW that has a simply closed boundary.

We start with the case of a degenerate pair of faces (f, g). Let w1,w2 be twovertices that are contained in different connected components of the intersection off and g. Let γf be a simple curve that lies in (intf ) ∪ w1,w2 and connects w1

and w2, where intf := f \ ⋃e ∈ E | e ∈ f . Similarly, let γg be a simple curvein (intg) ∪ w1,w2 connecting w1 and w2. Composing these curves, we obtain asimply closed curve γ which divides the plane by Jordan’s curve theorem into abounded and an unbounded component. We denote the bounded component by B .


Fig. 3 The right hand side shows an enumeration of the boundary walk around GW illustrated on the lefthand side. To isolate the subgraph G1, we pick the simply closed subwalk from w3 to w3

The set W = V ∩ B can be connected to V \ W only by walks which meet w1 or w2.For an illustration see the left hand side of Fig. 2.

We now turn to the case when there is no degenerate pair of faces. As D(F) = ∅,there must be a degenerate face f . Let w1 be a vertex in f with |(w1, f )| ≥ 2. Wepick an open simply connected neighborhood U of w1 that contains no other vertexexcept for w1 and e ∩ U is connected for all edges e adjacent to w1. Let x1 andx2 be two arbitrary points in two different connected components of (intf ) ∩ U . Weconnect x1 and x2 by a simple curve in intf and connect x1, x2 and w1 also by simplecurves that lie in the corresponding connected component of U . By composition, weobtain a simply closed curved and by Jordan’s curve theorem we get a bounded set B .Note that W = V ∩ B can be connected to V \ W only by walks which meet w1. Foran illustration see the right hand side of Fig. 2.

In both cases, we identified a set W which is finite and connected since thegraph G is locally finite and connected. Now, we find a subset V1 ⊆ W such thatG1 = (V1,E1) is bounded by a simply closed path. By construction, w1 lies in an un-bounded face of FW , which we denote by f and which has a finite boundary walk. Westart from w1 walking around f . The walk might visit certain vertices several times.The vertex w1 is visited at least twice, since the walk is finite. We pick a subsequenceof the walk such that no vertex is met twice except for the starting and ending vertex,which we denote by v1. In Fig. 3, this is illustrated (with v1 = w3).

The sequence of edges forms a closed path and encloses a subgraph, which wedenote by G1 = (V1,E1). Note that V1 ⊆ V contains v1, which might be equal to w1and V1 also might contain w2, which we denote in this case by v2. Nevertheless, v1and v2 are the only vertices in V1 that can be connected to V \ V1 by edges of G.Thus, G1 has the desired properties and we finished the proof.

The next lemma is the main tool for the proof of Theorem 1. It shows that theexistence of degenerate faces or pairs implies the presence of positive curvature.

Lemma 2 (Copy and paste lemma) Let G = (V ,E,F ) be a simple, planar graphthat is connected, locally finite, contains no terminal vertices and all extended edgesare regular. If D(F) = ∅, then there are subsets F ′ ⊆ F and V ′ ⊆ V such thatκGF (F ′) > 0 and κG

V (V ′) > 0.

Proof The idea of the proof is easy to illustrate. We assume D(F) = ∅ and take thesubgraph G1 of Lemma 1. We make copies of G1, paste them along the boundary


Fig. 4 An illustration of the copy and paste procedure

and embed the resulting graph into S2. See Fig. 4. We then show that the curvatureof this graph compared to the curvature in G does not increase, as long as we makeenough copies of G1. The statement is then implied by the Gauss–Bonnet formula.

Assume D(F) = ∅. By Lemma 1, there is a finite subgraph G1 = (V1,E1,F1) ofG which is enclosed by a closed path p. The vertex degree in G1 differs from G inat most two vertices, which we denote by v1 and v2. (If there is only one vertex, thenwe choose another vertex arbitrarily in the boundary path of G1.) Let e1, . . . , en bethe edges of the boundary path of G1 starting and ending at v1, i.e., v1 = e1 ∩ en.Moreover, let m < n such that v2 = em ∩ em+1. Denote p1 = e1, . . . , em andp2 = em+1, . . . , en. We take two copies G

(1)1 and G

(2)1 of G1. We paste them


along the edges of the subpaths p(1)2 and p

(2)2 (where p

(j)

2 corresponds to p2 in

G(j)

1 for j = 1,2), i.e., we identify the edges e(1)m+1, . . . , e

(1)n in G

(1)1 with the edges

e(2)m+1, . . . , e

(2)n in G

(2)1 . We denote the resulting graph by G2. Note that the edges

of the boundary path of G2 are the set p(1)1 ∪ p

(2)1 = e(1)

1 , . . . , e(1)m , e

(2)1 , . . . , e

(2)m .

Denote q1 = p(1)1 and q2 = p

(2)1 . Now, let N be an integer, that will be quantified

later. We take N copies G(1)2 , . . . ,G

(N)2 of G2 and paste them along the subpaths q

(j)

2

and q(j+1)

1 for j = 1, . . . ,N − 1. We embed the resulting planar graph into S2 and

we finally paste q(1)1 and q

(N)2 . We denote the resulting graph embedded in S2 by

G3 = (V3,E3,F3). In Fig. 4, the procedure is illustrated.We will make the following observations which are implied by the construction:

(O1) We can identify each corner in C(G3) uniquely with a corner in C(G) andC(G1) and they have the same multiplicity. For each such corner in C(G) orC(G1) there are exactly 2N corresponding corners in C(G3).

(O2) We can identify each face in F3 uniquely with a face in F and F1. Each boundedface in F1 can be identified uniquely with a face in F and exactly 2N corre-sponding faces in F3. Moreover, the face degree of corresponding faces is thesame in G, G1 and G3.

We now quantify N, which was introduced above. Let

N = maxj=1,2

⌈3|vj |

2(|vj |1 − 1)

⌉

,

where x denotes the smallest integer that is greater than x for x ∈ Q and | · |1denotes the vertex degree in G1. There is one more observation:

(O3) We can identify each vertex in V3 uniquely with a vertex in V and in V1. Eachof these vertices in V and V1 either corresponds to v1, v2 or to exactly N

vertices of the former boundary path p of G1 or to exactly 2N vertices in V3.Moreover, the vertex degree of a vertex in G3 is at least the vertex degree ofthe corresponding vertex in G and it is at least three.

The statement about the corresponding vertices in (O3) follows by construction. Tocheck the statement about the vertex degrees in (O3), one has to consider three cases.The statement is clear for vertices in V3 that are not contained in the boundary pathof G1. For the vertices v1, v2 the statement follows by the choice of N . For any othervertex v ∈ V3, the vertex degree |v|3 in G3 is equal to

∑c∈Cv(G3)

|c| = 2(|v| − 1),which yields the statement. Finally, we have |v|3 ≥ 3 for all v ∈ V3 since we assumedthat all extended edges in G are regular and all faces contained in F3 are bounded.

In the following, we will not distinguish between the objects we identified in (O1),(O2), (O3). An obvious consequence of (O1), (O2) and (O3) is that for a cornerc ∈ C(G3) and the corresponding corner c ∈ C(G) we have

κG3C (c) ≤ κG

C (c).

We now find F ′ ⊆ F that satisfies κGF (F ′) > 0. Denote the unbounded face in F1

by f∞ and set F ′ = F1 \ f∞, which we can identify with a subset of F and 2N


identical subsets of F3 by (O2). Thus, we conclude by the Gauss–Bonnet formulaand the inequality above that

2 = κG3V (V3) =

∑

c∈C(G3)

|c|κG3C (c) = 2N

∑

f ∈F ′

∑

c∈Cf (G3)

|c|κG3C (c)

≤ 2N∑

f ∈F ′

∑

c∈Cf (G)

|c|κGC (c) = 2N

∑

f ∈F ′κGF (f ).

Therefore, κGF (F ′) ≥ 1/N > 0.

To find V ′ ⊆ V with κGV (V ′) > 0, we assume that all vertices v ∈ V that corre-

spond to a vertex in the boundary path of G1 satisfy κGV (v) ≤ 0. Otherwise, we have

κGV (V ′) > 0 for V ′ = v. We first show κ

G3V (v) ≤ κG

V (v) for v ∈ V3. To do so, wecheck three cases: Firstly, the statement is clear for vertices v that are not containedin the boundary path p of G1. Secondly, for v ∈ V3 \ v1, v2 that is contained in theboundary path of G1, we have |v|3 = 2(|v| − 1). Hence,

κG3V (v) = 1 − |v|3

2+

∑

(v,f )∈Cv(G3)

∣∣(v, f )

∣∣ 1

|f |

= 2 − |v| +∑

(v,f )∈Cv(G)

2∣∣(v, f )

∣∣ 1

|f | − 2

|g| = 2κGV (v) − 2

|g| ≤ κGV (v),

where g ∈ F is the unique face adjacent to v in G for which there is no correspond-ing face in G3. Note that g corresponds to f∞ in G1 and |(v, g)| = 1. The secondequality follows since for any other corner in Cv(G) there are exactly two corre-sponding corners with the same multiplicity in Cv(G3). The last inequality is due tothe assumption κG

V (v) ≤ 0 for vertices v contained in p. Thirdly, we have to check

κG3V (vj ) ≤ κG

V (vj ) for j = 1,2. In the following inequality, we estimate |f | ≥ 3,then use

∑c∈Cvj

(G3)|c| = |vj |3 = 2N(|vj |1 − 1) ≥ 3|vj | from the definition of N , to

obtain

κG3V (vj ) = 1 − |vj |3

2+

∑

(vj ,f )∈Cvj(G3)

∣∣(vj , f )

∣∣ 1

|f | ≤ 1 − |vj |36

≤ 1 − |vj |2

.

As κGV (vj ) ≥ 1 − |vj |/2, we have κG

V (vj ) ≥ κG3V (vj ) for j = 1,2. Thus, we have

shown for all v ∈ V3

κG3V (v) ≤ κG

V (v).

We finish the proof by identifying a subset V ′ ⊆ V3 that satisfies κG3V (V ′) > 0 and,

by the identification (O3), κGV (V ′) > 0. Let V ′ ⊂ V be the vertices of G1 that are not

in the vertex set Vp of the boundary path p of G1. The Gauss–Bonnet formula and(O3) yield, since we assumed κG

V (v) ≤ 0 for v ∈ Vp

2 = κG3V (V3) = 2N

∑

v∈V ′κ

G3V (v)+N

∑

v∈Vp\v1,v2κ

G3V (v)+

∑

j=1,2

κG3V (vj ) ≤ 2NκG

V (V ′).


Therefore, κGV (V ′) ≥ 1

N> 0. If V ′ = ∅, then we are done. Otherwise, we arrived at

a contradiction to the assumption κGV (v) ≤ 0 for v ∈ Vp . In this case, we conclude

κGV (V ′′) > 0 for V ′′ = v for the vertex v that gives the contradiction.

The next lemma shows that the absence of degenerate faces and degenerate pairscharacterizes whether a “nice” graph is locally tessellating.

Lemma 3 Let G be a simple, planar graph that is connected, locally finite, containsno terminal vertices and each extended edge is regular. Then, G is locally tessellatingif and only if D(F) = ∅. In this case, each corner has multiplicity one and

κGV (v) = 1 − |v|

2+

∑

f ∈F,v∈f

1

|f | for all v ∈ V .

Proof Obviously, if D(F) is non-empty, then at least one of the conditions (T1),(T2∗), (T3∗) on p. 504 is violated. On the other hand, the absence of degenerate facesimplies that each corner has multiplicity one. This has the following consequences:Firstly, no edge can be included in only one face, which is (T1). Secondly, the bound-ary of each face is a path (i.e., there is a boundary walk of the face which is meetingevery vertex only once) and, hence, (T3∗) follows. Thirdly, the formula for the curva-ture now follows from the definition. Finally, the absence of degenerate pairs of facesimplies (T2∗) as extended edges are assumed to be regular.

Next, we show that non-positive curvature implies that the graph is “nice”. Theproof follows from straightforward calculation.

Lemma 4 Let G be a planar graph that is connected and locally finite.

(1) If κC(G) ≤ 0, then G is simple, admits no terminal vertices and each extendededge is regular. If κC(G) < 0, then there are no extended edges.

(2) If κV (G) ≤ 0, then G admits no terminal vertices and each extended edge isregular. If κV (G) < 0, then there are no extended edges.

(3) If κF (G) ≤ 0, then G is simple.

We come now to the proof of Theorem 1.

Proof of Theorem 1 If G is finite, then by the Gauss–Bonnet formula, it must admitsome positive curvature.

Lemma 4 implies that the assumptions of Lemma 2 are satisfied. Hence, byLemma 2, we have D(F) = ∅ and, by Lemma 3, we obtain that G is locally tes-sellating. Moreover, by Lemma 4, in the case of κC(G) < 0 or κV (G) < 0 the graphG admits no extended edges and, thus, G is strictly locally tessellating.

4 Embedding of a Locally Tessellating Graph into a Tessellation

In this section, we construct an embedding of a locally tessellating graph into a tes-sellating supergraph, which leaves crucial properties of a subset of vertices invariant.


This embedding allows us to carry over many results for tessellations to planar graphsin the forthcoming sections. We start the section with an extension of a propositionof Higuchi [15], to planar graphs. Then, after presenting the construction, we willextract some important properties of the tessellating supergraph.

Proposition 2 Let G be a simple, planar graph that is connected and locally finite.If κG

V (v) < 0 for all v ∈ V , then κV (G) ≤ −1/1806. The supremum is achieved forvertices with degree 3 that have adjacent faces with degrees exactly 3, 7 and 43.

Proof By Theorem 1 every non-positively curved graph is locally tessellating. In[15, Proposition 2.1] it is shown that κG

V < 0 on V implies κGV ≤ −1/1806 on V

for tessellating graphs and that the maximum is achieved as is claimed above. Theproof consists of a list of all relevant cases. The list is ordered by the vertex degreen and a vector (l1, . . . , ln) where l1 ≤ · · · ≤ ln are the degrees of the faces adjacentto the vertex. Since we allow for unbounded faces, we have to check some additionalcases: For n ≥ 5, the curvature for (l1, . . . , l4,∞) with l1, . . . , l4 ≥ 3 is smaller orequal to −1/6. For n = 4, the curvature for (3,3,3,∞) is zero and for (l1, l2, l3,∞)

with l1, l2 ≥ 3 and l3 > 3 the curvature is smaller or equal to −1/12. For n = 3, thecurvature of (3,6,∞) and (4,4,∞) is zero, while the curvature of (l1, l2,∞) withl1 ≥ 3, l2 > 6 is smaller or equal to −1/42 and with l1 ≥ 4, l2 > 4 it is smaller orequal to −1/20.

We now come to the construction of the embedding. Let G = (V ,E,F ) be a sim-ple, locally tessellating graph that satisfies κV (G) ≤ 0. Let W ⊆ V be a finite set ofvertices that is simply connected, i.e., both subgraphs GW and GV \W are connected.The construction consists of two steps. In the first step, we add binary trees to certainvertices. In the second step, we close the unbounded faces by adding “horizontal”edges.

Step 1: To every vertex v ∈ V with κGV (v) = 0 that is adjacent to n ≥ 1 infinigons,

we attach n binary trees. To do so, we embed every one of these trees into a differentinfinigon and then connect the roots of the trees and v by edges. With slight abuseof notation, we denote the face set of the resulting graph also by F .

Step 2: We choose the closing parameter, that is the size at which unbounded facesare closed by a “horizontal” edge. Let diam(W) := maxv,w∈W d(v,w). For ε > 0define

Rε := max

6,2 diam(W),(

2 + minv∈V

|v|)1

ε

.

By induction over n ∈ N, we perform the following procedure: For every unboundedface f ∈ ∂F Bn(W), we connect the two vertices in f ∩Sn(W) by an edge whenever|f ∩ Bn(W)| > Rε . (Note that the uniqueness of the two vertices follows since ex-tended edges are regular.) We denote, with slight abuse of notation, the face set ofthe modified graph after each induction step again by F .

This construction yields a graph that we denote by G′ = G′ε = (V ′,E′,F ′). Obvi-

ously, G′ is a supergraph of G, i.e., V ⊆ V ′, E ⊆ E′. Therefore, it is natural to talkabout corresponding vertices in V and V ′. In particular, we will denote for a vertex


v ∈ V the corresponding vertex in V ′ by v′ and for a subset W ⊆ V we denote thecorresponding subset in V ′ by W ′.

Theorem 2 Let G be a simple, connected, locally tessellating graph that satisfiesκV (G) ≤ 0, W ⊂ V be finite and simply connected and ε > 0. Then, the graph G′ =G′

ε constructed above is a tessellation and satisfies the following assertions:

(G1) If v ∈ W is not adjacent to an infinigon or κGV (v) < 0, then |v| = |v′|. Other-

wise, |v′| = |v| + n, where n is the number of adjacent infinigons. Moreover, ifκGC (v,f ) ≤ 0 for all (v, f ) ∈ Cv(G), then edges are added to v if and only if v

is the inner vertex of an extended edge.(G2) The embedding of G into the supergraph G′ is a graph isomorphism between

the subgraphs GW and GW ′ , (i.e., the adjacency relations of correspondingvertices W and W ′ remain unchanged). If κG

V < 0 on W , then the embedding iseven a graph isomorphism between the subgraphs GBG

1 (W) and GBG′

1 (W ′).

(G3) The distance of two vertices v,w ∈ W in G equals the distance of the corre-sponding vertices v′,w′ ∈ W ′ in G′.

(G4) If κC(G) ≤ 0, then κC(G′) ≤ min0, κC(G) + ε.(G5) If κV (G) ≤ 0, then κV (G′) ≤ min0, κV (G) + ε whenever ε ∈ (0,1/1806).

Proof It is obvious from the construction that G′ is a tessellating graph.(G1): Edges are added to vertices in W only in Step 1. This is exactly the case, if

the vertex v is adjacent to at least one infinigon and κGV (v) = 0. Then, as many edges

are added as there are adjacent infinigons. If κGC (v,f ) ≤ 0 for all (v, f ) ∈ Cv(G) and

κGV (v) = 0 then κG

C (v,f ) = 0 for all (v, f ) ∈ Cv(G). If v is adjacent to an infinigon,then |v| = 2 and both adjacent faces are infinigons.

(G2): The first statement follows since we do not connect or disconnect verticeswithin W . The second one follows from the first one and (G1).

(G3): Note that, in Step 2, we add edges and create paths in G′ that are not in G.By definition of Rε , such a path in G′ connecting vertices in W ′ has at least thelength diam(W). Therefore, the distance of v′,w′ ∈ W ′ in G′ is at least the distanceof v,w ∈ W in G. Moreover, by (G2) the distance does not increase either.

(G4): We consider three types of corners: Firstly, consider (v′, f ′) ∈ C(G′) that isthe corresponding corner of some (v, f ) ∈ C(G) with |f | < ∞. Clearly, κG

C (v,f ) ≤κG′C (v′, f ′) by (G1). Secondly, let (v′, g) be such that v′ is the corresponding vertex

of some vertex v ∈ V and g is created in Step 2 by closing an infinigon. Obviously,|v′| ≥ max3, |v| and |g| ≥ max6,1/ε by construction and the definition of Rε .Therefore, κG′

C (v′, g) ≤ min1/3,1/|v| − 1/2 + min1/6, ε ≤ min0, κC(G) + ε.Thirdly, let (w,g) ∈ C(G′) be a corner of a vertex w that was added with a binary treein Step 1. In this case, κC(G) = 0. Obviously, |w| ≥ 3 by construction and |g| ≥ 6 bydefinition Rε . Therefore, κG′

C (w,g) ≤ 1/3 − 1/2 + 1/6 = 0 = κC(G).

(G5): We consider three cases: Firstly, let v′ ∈ V ′ be the corresponding vertex ofsome vertex v to which n ≥ 1 binary trees were added in Step 1. Then, n = |v′| − |v|by (G1). This gives a total of 2n new faces g ∈ F ′, each of which has face degree


|g| ≥ 6 after being closed in Step 2. Therefore,

κG′V (v′) ≤ 1 − |v′|

2+

∑

f ∈F,v∈f,|f |<∞

1

|f | + |v′| − |v|3

= κGV (v) − |v′| − |v|

6≤ κG

V (v).

Secondly, let v′ ∈ V ′ be the corresponding vertex of some v ∈ V to which no edgeswere added in Step 1. If v is not adjacent to an infinigon, then no edges are addedin Step 2 either. Thus, κG

V (v) = κG′V (v′). Otherwise, κG

V (v) < 0 (as, otherwise, edgeswere added in Step 1). Due to planarity, at most two edges were added to v in Step 2,i.e., |v| ≤ |v′| ≤ |v| + 2. Moreover, for a face g ∈ F ′ adjacent to v′, there is eithera corresponding face f ∈ F and |g| = |f | or g was created in Step 2 from an in-finigon, in which case |g| ≥ (minv∈V |v| + 2)/ε =: 1/δ by definition of Rε . By theseconsiderations, we get

κG′V (v′) ≤ 1 − |v|

2+

∑

f ∈F,v∈f

1

|f | + |v′|δ ≤ κGV (v) + (|v| + 2)δ ≤ κG

V (v) + ε,

where the last inequality follows by the definition of δ. We have κG′V (v′) < 0 as

κGV (v) ≤ −ε by Proposition 2 whenever ε ∈ (0,1/1806). Thirdly, let w ∈ V ′ be a

vertex that has no corresponding vertex in V , i.e., it is a vertex of a binary tree whichwas added in Step 1. Note that, in this case, κV (G) = 0 and |w| ≥ 3 by Step 1 andStep 2. Moreover, by definition of Rε , all faces g ∈ F ′ adjacent to w satisfy |g| ≥ 6.We get κG′

V (w) ≤ 1 − |w|/2 + |w|/6 ≤ 0 = κV (G).

5 Geometric Applications

In this section, we discuss some applications of the fact that every non-positivelycurved planar graph is locally tessellating. Indeed, by the embedding constructed inthe previous section, most of statements for locally tessellating graphs are now directconsequences of the results for tessellations. These results concern the absence of cutlocus, the boundary structure of distance balls, estimates for the growth of distanceballs, bounds and positivity of Cheeger’s constant and empty interior of minimalbigons.

5.1 Absence of Cut Locus

The cut locus of a vertex v0 of a graph is the set of all vertices, where the distancefunction d(v0, ·) attains a local maxima. In contrary, empty cut locus for all verticesimplies that geodesics can be continued ad infinitum.

For non-positively curved tessellation, a corresponding result can be found in [3].Note that in [2, 3] the results and proofs are given for the metric space consideringthe distance function on the faces of the graph. However, the results are true for themetric space of vertices as well, since the dual graph of a non-positively corner curvedtessellation is again a non-positively corner curved tessellation.


Theorem 3 Let G = (V ,E,F ) be a planar graph that is connected, locally finiteand satisfies κC(G) ≤ 0. Then, the metric space (V , d) has no cut locus.

Proof By Theorem 1, the graph G is locally tessellating. Suppose there is v0 ∈ V

with non-empty cut locus and suppose v ∈ V is in the cut locus of v0. Then, by thedefinition of the cut locus, all adjacent vertices of v have smaller or equal distanceto v0. Consider a simply connected set of vertices W that contains all vertices of thepaths of minimal length from v0 to v and all adjacent vertices of v. By (G2), (G3) ofTheorem 2, there is a tessellation G′ such that the distances of corresponding verticesin W and W ′ agree. Let v′

0, v′ ∈ W ′ be the corresponding vertices to v0, v in W . Since

κC(G) ≤ 0 and since v cannot be the inner vertex of an extended edge, no edges areadded to v by (G1). Therefore, v′ is in the cut locus of v′

0 in G′. By (G4), we haveκC(G′) ≤ 0. This leads to a contradiction to [2, Theorem 1], which guarantees ab-sence of a cut locus for tessellations under the assumption of non-positive cornercurvature.

5.2 The Boundary of Distance Balls

Non-positive corner curvature has very strong implications on the boundary structureof distance balls. In particular, the concept of admissibility introduced in [2, 3] cap-tures important aspects of the boundary behavior. Since this concept is quite involved,we only derive some of its most important consequences.

In [30], some of these statements were already proven under various assump-tions, which all imply κC(G) < 0. In particular, these statements are used there toprove positivity of Cheeger’s constant. Here, we will use these properties to proveabsence of finitely supported eigenfunctions of nearest neighbor operators on planarnon-positively curved graphs.

Theorem 4 Let G be a planar graph that is locally finite and satisfies κC(G) ≤ 0.Let v0 ∈ V , n ∈ N0 and denote Bn := Bn(v0) and Sn := Sn(v0).

(1) Every vertex in Sn is adjacent to at least one vertex in Sn+1.(2) Every vertex in Sn+1 is adjacent to at most two vertices in Sn.(3) If two vertices in Sn have a common neighbor in Sn+1, then both of them have

another neighbor in Sn+1.(4) Let f1, . . . , f2k be a cyclic enumeration of the faces of ∂F Bn. Then, the case

|f2j−1 ∩ Bn| = |f2j ∩ V \ Bn| = 1 for all 1 ≤ j ≤ k cannot happen.(5) The sphere Sn admits a cyclic enumeration in the sense that two succeeding ver-

tices are adjacent to a common boundary face in ∂F Bn.

Proof Let G′ be the tessellating graph constructed from Bn+1 in G. As the embed-ding does not change distances of vertices in Bn+1, by (G3). The spheres Sn,Sn+1 ⊂V can be considered as subsets of the spheres S′

n := Sn(v′0) and S′

n+1 := Sn+1(v′0)

in G′.(1) Suppose v ∈ Sn is not adjacent to any vertex in Sn+1. Then, v is in the cut locus

of v0, which is a contradiction to Theorem 3.


(2) Suppose v ∈ Sn+1 is adjacent to more than two vertices in Sn. Then, the cor-responding vertex v′ of v in G′ is connected to more than two vertices in S′

n+1. Thisgives a contradiction to [3, Proposition 2.5(a)] (where a corresponding statement isfound for the dual graph).

(3) Let u,v ∈ Sn be adjacent to some w ∈ Sn+1. Denote by u′, v′,w′ the corre-sponding vertices in V ′. The subgraph Gu,v,w is a path and is not included in anextended edge (otherwise, this leads to a contradiction to u,v ∈ Sn since every ex-tended edge is regular). Hence, there is a unique face f ∈ ∂F Bn that contains u,v,w

and this face is bounded. By the construction of G′, the face f has a correspond-ing boundary face f ′ in G′. By [3, Corollary 2.7], occurrence of such a face impliesthat the corresponding vertices u′, v′ ∈ S′

n have neighbors u′1, v

′1 ∈ S′

n+1 such thatu′

1, v′1 = w′. (In the language of [3], the dual vertex of the face f has label b and,

therefore, its neighbors in the boundary have label a+ by the admissibility of dis-tance balls. Translating this to our situation, we obtain the conclusion above.) SinceκC(G) ≤ 0 and the subgraph Gu,v,w is not included in an extended edge, no edgeswere added to the vertices u,v in the construction of G′, by (G1). Therefore, thereare u1, v1 = w in Sn+1 whose corresponding vertices are u′

1, v′1.

(4) Assume the opposite. Then, Bn+1 encloses no infinigon. By (G1), no edgeswere added to Bn+1 while embedding it into a tessellation. The corresponding state-ment for tessellations, [21, Proposition 13], now gives a contradiction.

(5) We have κC(G′) ≤ 0, by (G4). Thus, S′n admits a cyclic enumeration, by [2,

Theorem 3.2]. Since Sn can be considered as a subset of S′n, this gives enumeration

of Sn. Now, it can be easily seen that this is a cyclic enumeration of Sn.

5.3 Growth of Distance Balls

In this subsection, we give estimates for the exponential growth of distance balls interms of curvature. A lower bound for tessellations is found in [2, Theorem 5.1] andan upper bound in [20, Theorem 4].

Theorem 5 Let G be a simple, planar graph that is connected, locally finite, has nocut locus and satisfies κV (G) < 0. Then, for all v ∈ V and n ≥ 1

∣∣Sn(v)

∣∣ ≥ −2κV (G)

q

q − 1

∣∣Bn−1(v)

∣∣,

where q := supf ∈F |f | and q/(q − 1) = 1 in the case q = ∞. Moreover, for the

exponential growth rate μ := lim supn→∞ 1n

log |Sn(v)| and p := supv∈V |v|, one has

log

(

1 − 2κV (G)q

q − 1

)

≤ μ ≤ log(p − 1).

Remark By Lemma 4(1) and Theorem 3, the assumptions that G is simple and hasno cut locus is implied by κC(G) ≤ 0.

Proof of Theorem 5 Theorem 1 implies that G is strictly locally tessellating. By thenegative vertex curvature, (G2) and (G3) imply that the distance-n-ball of a vertex


in G is isomorphic to the distance-n-ball for the corresponding vertex in a tessellat-ing graph G′. Therefore, |SG

n (v)| = |SG′n (v′)| and |BG

n (v)| = |BG′n (v′)|. By (G5), we

have κV (G′) ≤ κV (G)+ ε for ε ∈ (0,1/1806). Combining this with the statement fortessellations, [2, Theorem 5.1], we obtain

∣∣SG

n (v)∣∣ = ∣

∣SG′n (v′)

∣∣ ≥ −2κV (G′) q

q − 1

∣∣BG′

n−1(v′)∣∣

≥ −2(κV (G) + ε)q

q − 1

∣∣BG

n−1(v)∣∣.

As ε can be chosen arbitrarily small, we obtain the first result. The lower bound in thesecond statement is a direct consequence of the first one (compare [2, Corollary 5.2]).The upper bound follows from a comparison to a p-regular tree. For more details see[20, Theorem 4].

5.4 Estimates for the Cheeger Constant

An isoperimetric constant known as the Cheeger constant plays an important role inmany areas of geometry, probability and spectral theory. For infinite graphs, it wasfirst defined by Dodziuk [6] and later in another version by Dodziuk/Kendall [8].These different versions appear in connection to different versions of the discreteLaplace operator. In [6, 8], the respective constant is used to estimate the bottom ofthe spectrum. In probability, positivity of Cheeger’s constant implies that the sim-ple random walk is transient. This and various other implications can be found in[14, 27, 28].

For a subset U ⊆ V , let the Cheeger constants be defined as

αU := inf

|∂EW |vol(W)

∣∣∣∣W ⊆ Ufinite

and α := αV ,

βU := inf

|∂EW ||W |

∣∣∣∣W ⊆ Ufinite

and β := βV ,

where ∂EW is the set of edges that connect a vertex in W with a vertex in V \ W andvol(W) = ∑

v∈W |v|. The constant α was first introduced in [8] and β in [6]. The setK of finite subsets of V forms a net with respect to the inclusion relation. We definethe following limits along this net

α∂V := limK∈K

αV \K and β∂V := limK∈K

βV \K.

In [12] the quantity α∂V was introduced as α∞.Woess [27] and Zuk [30] proved separately that negative curvature implies a strong

isoperimetric inequality, i.e., have positive Cheeger constant. While [27] assumesthat the graph is tessellating and an average curvature is negative, [30] allows forinfinigons and his assumptions imply negative corner or face curvature. Positivity ofCheeger’s constant was also proven later by Higuchi [15] under the stronger assump-tion of negative vertex curvature in the case of tessellations. Explicit formulas for theCheeger constant of regular tessellations can be found [16, 17]. In [20] lower boundsfor both types of Cheeger’s constant are obtained in the context of locally tessellating


graphs in terms of curvature. Moreover, Fujiwara [12] proved that the Cheeger con-stant at infinity α∂V is equal to one for trees with vertex degree (and hence curvature)tending to negative infinity. In [19], it is shown that this implication holds also fortessellating graphs.

Theorem 6 Let G be a simple, planar graph that is connected, locally finite andsatisfies κV (G) ≤ 0.

(1) For all U ⊆ V , we have

αU ≥ 1 − 1

pU

2qU

qU − 2and βU ≥ pU − 2qU

qU − 2,

where pU := infv∈U |v|, qU := inff ∈F,f ∩U =∅ |f | and the conventions 1/∞ = 0,∞/∞ = 1.

(2) If limK∈K supv∈V \K κGV (v) = −∞, then α∂V = 1 and β∂V = ∞.

(3) We have

α ≥ −2C supv∈V

1

|v|κGV (v) and β ≥ −2CκV (G),

where C := (1 + 2Q−2 )(1 + 2

(P−2)(Q−2)−2 ), with P := supv∈V |v|,Q := supf ∈F |f | and the conventions 1/∞ = 1/(0 · ∞ − 2) = 0.

(4) β ≥ α > 0, whenever κGV < 0 on V .

Remark (a) Let γ := (q − 2)/2q be the 2π -normalized angle of a regular q-gon andκ := 1 − pγ be the averaged curvature. We can reformulate the estimates of (1) forU = V in terms of curvature as follows

α ≥ −κ

1 − κand β ≥ − κ

γ.

While these estimates are new, (2) is an extension of [12, 19], (3) is an extension of[20] and (4) is a version of [15, 27, 30]. Note also that the first inequality of (4) isindependent of the assumption κG

V < 0.(b) For the proof of (1) and (2), we do not need the embedding of Sect. 4. This is

important as pU and qU might be changed by Step 1 and Step 2.

Proof of Theorem 6 By Theorem 1, the graph G is strictly locally tessellating.(1) We claim that

|∂EW | ≥ vol(W) − 2qU

qU − 2

(|W | + c(W) − 2),

where c(W) is the number of connected components of GU\W . For the proof of theformula, we follow the lines of the proof of [19, Lemma 1], only instead of using theestimate |f | ≥ 3, we go with |f | ≥ qU for all faces f ∈ F with |f ∩ U | = ∅. (Com-pare also to Proposition 2.2 and 2.3 in [20].) We obtain for all finite and connected


sets W ⊆ U with c(W) ≤ 2

|∂EW |vol(W)

≥ 1 − 2qU

qU − 2

|W |vol(W)

≥ 1 − 1

pU

2qU

qU − 2,

|∂EW ||W | ≥ vol(W)

|W | − 2qU

qU − 2≥ pU − 2qU

qU − 2.

By the same arguments as in the proof of Proposition 6 in [19], it suffices toconsider finite, connected sets W ⊆ U with c(W) ≤ 2. Thus, the formulas aboveyield (1).

(2) By the formula for the curvature of Lemma 3, we have supv∈V \K κGV (v) →

−∞ if and only if pV \K → ∞ along the net K ∈ K. Hence, (2) follows from (1).(3) Theorem 1 of [20] states (3) for locally tessellating G under the assumptions

that the right hand sides are positive. By Proposition 2, κV (G) < 0 is implied byκGV < 0 on V . Moreover, one checks that κG

V (v)/|v| ≤ 1/|v| − 1/2 + 1/3 ≤ −1/42for |v| > 6 since |f | ≥ 3 for all f ∈ F . Therefore, also supv∈V κG

V (v)/|v| < 0 ifκGV < 0 on V .

(4) The statement follows directly from (3), as the right hand side is positive bythe considerations in the proof of (3).

5.5 Empty Interior of Minimal Bigons

In this section, we discuss a geometric property that is related to hyperbolicity.In [23], it is shown for Cayley graphs of discrete groups that empty interior of mini-mal bigons is equivalent to Gromov hyperbolicity. Since we allow for arbitrary largefaces, this equivalence is not true in our context. However, in [30, Corollary 1], Gro-mov hyperbolicity is shown by proving empty interior of minimal bigons under vari-ous assumptions implying negative corner curvature and the assumption of a uniformbound on the degree of polygons, see also [3, Theorem 2]. Despite of that, if only allminimal bigons have empty interior, then one still can construct the Floyd-boundaryof G and show that it is homeomorphic to S1. For a detailed discussion and referencessee [18].

Let us introduce the notion of a minimal bigon. Let p1 = (v1, . . . , vn) and p2 =(w1, . . . ,wn) be the vertices of two finite paths satisfying d(v1, vn) = d(w1,wn) = n

and v1 = w1, vn = wn. Such a pair (p1,p2) is called a bigon. A bigon is calledminimal if vj = wj for j = 1, n. The interior of a minimal bigon are all verticesenclosed by the two paths that do not belong to any of them.

Theorem 7 Let G be a planar graph that is connected, locally finite and satisfiesκC(G) < 0. Then, any minimal bigon has empty interior. Moreover, if there is uniformupper bound on the face degree of the polygons, then the graph is Gromov hyperbolic.

Remark Note that the assumption for the Gromov hyperbolicity only excludes theexistence of arbitrary large polygons but not the existence of infinigons.

Proof Since κC(G) < 0, the graph is strictly locally tessellating by Theorem 1. LetW be the union of the vertices of a bigon and its interior. Obviously, W is sim-


ply connected. Moreover, note that κC(G) < 0 implies κV (G) < 0. By (G2), (G3),(G4) there is a tessellation G′ with κC(G′) < 0 such that the distance of verticesin W remain unchanged compared to the corresponding set W ′ in G′. Thus, W ′ isa minimal bigon as well. By [3, Theorem 2], any minimal bigon in G′ has emptyinterior and the statement follows from (G2). The statement about the Gromov hy-perbolicity now follows from the arguments of [23] (compare to [30, Corollary 1] and[3, Corollary 5]).

6 Applications in Spectral Theory

We start this section by introducing two well known versions of the discrete Laplaceoperator. Then, we recall the corresponding bounds for the bottom of the spectrumimplied by the Cheeger constant. After that, we show that the essential spectrum forboth versions of the Laplacian is trivial, whenever the curvature decreases uniformlyto −∞. This is an extension of [12, 19]. Finally, extending [21], we prove that nearestneighbor operators on non-positively corner curved graphs have no finitely supportedeigenfunctions.

Let c(V ) be the space of complex valued functions on V and cc(V ) the space offunctions that are zero outside a finite set. The discrete Laplace operator , oftenused in mathematical physics, is acting as

(ϕ)(v) =∑

w∼v

(ϕ(v) − ϕ(w)

)

and is essentially self-adjoint on cc(V ), (for a proof see [29]). We denote the self-adjoint extension on 2(V ) = ϕ ∈ c(V ) | ∑

v∈V |ϕ(v)|2 < ∞ also by . Anotherversion of the Laplacian , often used in discrete spectral geometry, acting as

(ϕ)(v) = 1

|v|∑

w∼v

(ϕ(v) − ϕ(w)

)

on 2(V , | · |) = ϕ ∈ c(V ) | ∑v |v||ϕ(v)|2 < ∞ is a bounded, self-adjoint operator.

6.1 Spectral Bounds and Triviality of Essential Spectrum

The following bound for the bottom of the spectrum of can be derived from [22],see also [1, 12]

1 −√

1 − α2 ≤ infσ().

Here, α is the Cheeger constant defined in Sect. 5.4. This extends to a bound for thebottom of the spectrum of the Laplacian (see [19], compare also [29])

(1 −

√1 − α2

)infv∈V

|v| ≤ infσ().

Hence, the bounds on α in Theorem 6 give bounds for the bottom of the spectrum.The next theorem generalizes a result in [19], see also [12].


Theorem 8 Let G be a simple, planar graph that is connected, locally finite andsatisfies κV (G) ≤ 0. Then

(1) σess() = 1 follows if κGV (vn) → −∞ for vn → ∞,

(2) σess() = ∅ if and only if κGV (vn) → −∞ for vn → ∞,

where the limit vn → ∞ means that the sequence eventually leaves every finite set.

Proof Under the assumptions above, Theorem 6(2) yields α∂V = 1. This impliesσess() = 1 by [12, Theorem 1]. The equivalence in (2) follows from [19, The-orem 2] and the fact that the vertex degree tends to ∞ if and only if the curvaturetends to −∞.

6.2 Absence of Finitely Supported Eigenfunctions

A linear operator A defined on a subspace of c(V ) is called a nearest neighboroperator on G if its matrix representation in the standard basis is given by somea : V × V → C such that a(w,v) = 0 if v ∼ w and a(w,v) = 0 if v ∼ w and v = w.Hence, A acts as

(Aϕ)(v) =∑

w∈V

a(v,w)ϕ(w) = a(v, v)ϕ(v) +∑

v∼w

a(v,w)ϕ(w).

The operators and (possibly plus multiplication by a potential) are nearest neigh-bor operators. The following theorem is proven in [21] for tessellating graphs.

Theorem 9 Let G be a planar graph that is connected, locally finite and satisfiesκC(G) ≤ 0. Then, a nearest neighbor operator on G does not admit finitely supportedeigenfunctions.

The proof in [21] is based on an induction over the distance balls of the metricspace of faces. Since we allow for unbounded faces, these distance balls might haveinfinite cardinality. Therefore, the proof cannot be carried over directly. We will givean alternative proof which uses a representation of the operator in polar coordinatesand makes use of what we know about the boundary of distance balls, Theorem 4.

Let G = (V ,E,F ) be a planar graph that is locally finite and satisfies κC(G) ≤ 0.For v0 ∈ V , denote Sn = Sn(v0) and sn = |Sn|. By Theorem 4(5), we have a cyclicenumeration of Sn. We reorder the enumeration in the spheres inductively by cyclicpermutation. Let v

(n)1 be the first vertex in the enumeration of Sn. We shift the enu-

meration of Sn+1 such that in the new enumeration v(n+1)1 is the first vertex (with

respect to the unshifted enumeration of Sn+1) that is adjacent to v(n)1 . Inductively, we

get an enumeration, v(n)1 , . . . , v

(n)sn for all spheres Sn.

For a function ϕ ∈ c(V ), let ϕn be the restriction to c(Sn). For a nearest neighboroperator A, let the matrices En ∈ Csn+1×sn , Dn ∈ Csn×sn , E′

n ∈ Csn×sn+1 be givensuch that

(Aϕ)n = −En−1ϕn−1 + Dnϕn − E′nϕn+1.


Then, Dn is the restriction of A to c(Sn) and the matrices En and E′n are given

by En(i, j) = a(v(n+1)i , v

(n)j ) and E′

n(j, i) = a(v(n)j , v

(n+1)i ) for i = 1, . . . , sn+1, j =

1, . . . , sn. A similar construction was given in [10].

Lemma 5 Let G be a planar graph that is connected, locally finite and satisfiesκC(G) ≤ 0. Then, for n ∈ N0, we have the following:

(1) Each column of En has at least one non-zero entry.(2) Each row of En has exactly one or two non-zero entries. Two non-zero entries

always correspond to succeeding vertices in the enumeration of Sn.(3) Each two columns of En have at most one non-zero entry at the same compo-

nent. In this case, each of the columns have another non-zero entry at a distinctcomponent.

Proof Statement (1) follows since each vertex in Sn is connected to a vertex in Sn+1by Theorem 4(1). Statement (2) follows since each vertex in Sn+1 is connected to atmost two vertices in Sn, by Theorem 4(2). The other statement of (2) follows fromthe enumeration of the distances spheres. Statement (3) follows from the planarityand Theorem 4(3).

Lemma 6 Let G be a planar graph that is connected, locally finite and satisfiesκC(G) ≤ 0. Then, En is injective for all n ∈ N0.

Proof Suppose En is not injective, i.e., its column vectors are linearly dependent. Bythe preceding lemma, En must be of the form

En =

⎛

⎜⎜⎜⎝

an(1,1) an(1, sn)

an(2,1). . .

. . . an(sn+1 − 1, sn − 1)

an(sn+1, sn − 1) an(sn+1, sn)

⎞

⎟⎟⎟⎠

,

where the entries an(i, j) = a(v(n+1)i , v

(n)j ) are non-zero, while all other entries are

zero. However, this situation is geometrically impossible by Theorem 4(4).

Proof of Theorem 9 Suppose ϕ ∈ cc(V ) is an eigenfunction of A to λ ∈ C. Let v0 ∈ V

be such that ϕ(v0) = 0 and k ∈ N such that ϕk−1 ≡ 0 and ϕn ≡ 0 for n ≥ k. Rewritingthe eigenvalue equation (Aϕ)k = λϕk on the kth sphere, one has

Ek−1ϕk−1 = (Dk − λ)ϕk − E′kϕk+1.

By the choice of k, the right hand side is equal to zero. Since Ek−1 is injective byLemma 6 and ϕk−1 ≡ 0, the left hand is non-zero. This is a contradiction.

Acknowledgements The author would like to express his gratitude to Daniel Lenz for pointing outsome of the questions that inspired this research. Moreover, many valuable remarks and suggestions weregiven by Norbert Peyerimhoff on an earlier version of this paper. The author is also thankful to the anony-mous referee for several hints on the literature. Furthermore, he acknowledges the financial support by theGerman Science Foundation (DFG) and the Klaus Murmann Fellowship Programme (SDW).


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2. Baues, O., Peyerimhoff, N.: Curvature and geometry of tessellating plane graphs. Discrete Comput.Geom. 25, 141–159 (2001)

3. Baues, O., Peyerimhoff, N.: Geodesics in non-positively curved plane tessellations. Adv. Geom. 6(2),243–263 (2006)

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Zusammenfassung in deutscher Sprache

In dieser Arbeit studieren wir den Einfluß der Geometrie eines Rau-mes auf die spektralen und stochastischen Eigenschaften von Laplace-operatoren, wie sie in verschiedenen mathematischer Disziplinen be-trachtet werden. In der Tat sieht man bei der Untersuchung von Rie-mannschen Mannigfaltigkeiten und Graphen eine Vielzahl an Gemein-samkeiten. Auf der anderen Seite gibt es in der Riemannschen Ge-ometrie sehr naturliche Begriffe wie Abstand und Krummung, die sichkanonisch aus der Geometrie ergeben, und die jedoch keine eindeuti-gen Analogien im Diskreten haben. Bei einer ersten naheliegendenAbstandsdefinition im Diskreten ist sogar ein Auseinanderklaffen bes-timmter Phanomene im Vergleich zum Kontinuierlichen zu beobachten.Die Betrachtung und Untersuchung geeigneter diskreter Abstands- undKrummungsbegriffe ist deshalb ein Leitmotiv dieser Arbeit.

Eine vereinheitlichende Perspektive fur Laplaceoperatoren auf Man-nigfaltigkeiten und Graphen bildet die Theorie der Dirichletformen.Aus einer physikalischen Perspektive konnen Dirichletformen als “En-ergiefunktionale” verstanden werden. In dieser Arbeit beschranken wiruns darauf Dirichletformen auf diskreten Raumen zu betrachten. DerVorteil dieser Raume ist, dass sie oft eine sehr explizite und wenig tech-nische Behandlung erlauben. Trotzdem ist unsere Prasentation daraufbedacht den Weg zum Studium Dirichletformen auf allgemeinen Raumezu ebenen.

Im ersten Kapitel fuhren wir den Rahmen und die grundlegendenKonzepte ein. Das beinhaltet als erstes eine Eins-zu-eins-Beziehungzwischen (gewichteten) Graphen und regularen Dirichletformen aufdiskreten Raumen. Mittels dieser Formen definieren wir dann Laplace-operatoren und ihre Halbgruppen und diskutieren erste Eigenschaftenund wichtige Beispiele. Zweitens betrachten wir die Warmeleitungsglei-chung und das Konzept der stochastischen Vollstandigkeit, welcheszur Charakterisierung der Eindeutigkeit beschrankter Losungen dient.Die Originalmanuskripte, auf denen diese ersten beiden Abschnitteberuhen, sind [KL12, KL10], welche im zweiten Teil der Arbeit zufinden sind. Im weiteren Verlauf des Kapitels werden drei Klassen vonBeispielen diskutiert. Die erste Klasse sind Graphen uber diskretenMaßraumen, deren Maß nach unten durch eine positive Konstante

357

358 ZUSAMMENFASSUNG IN DEUTSCHER SPRACHE

beschrankt ist. Die dort prasentierten Resultate basieren auf den Orig-inalmanuskripten [KL12, BHK13]. Zweitens betrachten wir Graphenmit einer schwachen Form spharischer Symmetrie, wie sie in dem Origi-nalmanuskript [KLW13] zu finden sind, und studieren Symmetrien desWarmekerns, spektrale Schranken und stochastische Vollstandigkeit.Letztendlich werden noch Resultate fur dunne Graphen, d.h. Graphenmit relativ wenig Kanten, prasentiert, welche den Originalmanuskripten[BHK13, KL10] entnommen sind.

Im zweiten Kapitel studieren wir Abstand und daraus abgeleiteteBegriffe wie Volumenwachstum. An dieser Stelle treten im Vergleich zuRiemannschen Mannifaltigkeiten klare Unterschiede zu Tage wenn manden naiven Zugang uber die kombinatorische Graphenmetrik wahlt.Dies wurde zuerst von Wojciechowski [Woj08, Woj11] beobachtetund findet sich auch in [CdVTHT11a, KLW13] wieder. Diese Un-terschiede treten fur unbeschrankte Operatoren auf Graphen auf. Aufder anderen Seite liefert der normalisierte Laplaceoperator, welcherimmer beschrankt ist, die Ergebnisse, die die Riemannsche Betrach-tungsweise erwarten lasst. Aus diesem Blickwinkel ist die kombina-torische Graphenmetrik die naturliche Wahl fur den normalisiertenLaplaceoperator, wahrend sie sich fur die Betrachtung unbeschrank-ter Laplaceoperatoren nicht eignet. Dies legt nahe fur einen gegebe-nen Operator jeweils nach einem geeigneten Abstandsbegriff Ausschauzu halten. Diese Herangehensweise hat sich bereits als außerst effek-tiv im Rahmen stark lokaler, regularer Dirichletformen herausgestellt.Fur solche Formen wurden sogenannte intrinsiche Metriken genutztum Ergebnisse fur Riemannsche Mannigfaltigkeiten sehr stark zu ve-rallgemeinern. Dies ist besonders in der grundlegenden systematis-chen Betrachtung von Sturm [Stu94] zu finden. Vor Kurzem wurdedieses Konzept intrinsicher Metriken fur allgemeine regulare Dirich-letformen durch Frank/Lenz/Wingert verallgemeinert [FLW14]. (EinVorabdruck dieser Arbeit war schon etwa 5 Jahre vor ihrer Veroffent-lichung im Umlauf.) In dieser Arbeit nutzen wir solche intrisischenMetriken um allgemeine Laplaceoperatoren auf Graphen zu studieren.Auf diese Weise zeigen wir Resultate, die als naturliche Entsprechun-gen ihrer Gegenstucke in der Riemannschen Geometrie erscheinen. EinHohepunkt dieses Kapitels ist eine sogenannte Cheeger Ungleichung,welche in [BKW14] bewiesen wurde. Eine solche Ungleichung imZusammenhang mit intrinsichen Metriken ist in einem gewissen Sinneuberraschend, da es auf den ersten Blick nicht offensichtlich ist, in-wiefern ein Abstandsbegriff bei der Definition einer isoperimetrischenKonstante eine Rolle spielt. In der Tat wird hier ein offenes Problemvon Dodziuk/Kendall [DK86] aus dem Jahre 1986 gelost. Weiterhinprasentieren wir sogenannte Liouville Theoreme von Yau and Karp, einTheorem von Gaffney uber wesentliche Selbstadjungiertheit, Brooks’and Sturm’s obere Schranken fur den unteren Rand des (wesentlichen)

ZUSAMMENFASSUNG IN DEUTSCHER SPRACHE 359

Spektrums und Sturm’s p-Unabhangigkeit der Spektren fur Graphen.Die Resultate gehen auf die Originalmanuskripte [HK14, HKMW13,HKW13, BHK13] zuruck. Alle diese Ergebnisse sind die ersten ihrerArt fur allgemeine Laplaceoperatoren auf Graphen. Sie beinhaltenden normalisierten Laplaceoperator als Spezialfall und verbessern dieErgebnisse fur diesen speziellen Operator sogar in einigen Fallen.

Im dritten und letzen Kapitel geht es um Krummung. Wir betra-chten hier ausschließlich planare Graphen mit Standardgewichten. Fursolche Graphen existiert ein sehr intuitiver Krummungsbegriff, welchersich bis hin zu Betrachtungen von Descartes zuruckverfolgen lasst. DerHauptfokus liegt auf spektralen Konsequenzen oberer Krummungs-schranken. Wir zeigen mittels Krummungsschranken obere und un-tere Schranken fur den unteren Rand des Spektrums welche [KP11]entnommen sind. Außerdem charakterisieren wir die Diskretheit desSpektrum mittels Krummung basierend auf [Kel10]. Das liefert einResultat, welches als Analogie zu einer Arbeit von Donnelly/Li [DL79]verstanden werden kann. Im Fall von rein diskretem Spektrum gebenwir außerdem Eigenwertasymptotiken an, was direkt aus den Resul-taten von [BGK13] folgt. Zum Abschluss wenden wir die Ergebnisseaus [BHK13] an, um Aufschluss uber die p-Unabhangigkeit des Spek-trums zu erhalten.

Lebenslauf

361

Curriculum Vitae – Matthias Keller

Curriculum Vitae

Kontaktdaten

Mathematisches InstitutErnst-Abbe-Platz 2Friedrich Schiller Universitat07743 JenaE-mail: [email protected]:analysis-lenz.uni-jena.de

Personliche Daten

Geburtsdatum: 31. Dezember 1980Staatsangehorigkeit: deutschFamilienstatus: verheiratet, drei Kinder

Ausbildung

Promotion in MathematikFriedrich-Schiller-Universitat Jena, On the spectral theory of operatorson trees, Gutachter: Daniel Lenz, Simone Warzel und Richard Froese,Dezember 2010.

Diplom in MathematikTechnische Universitat Chemnitz, Produkte zufalliger Matrizen undder Lyapunov-Exponent, Gutachter: Peter Stollmann und Daniel Lenz,Juni 2006, mit Nebenfach Physik, Vordiplom in Wirtschaftsmathema-tik.

Werdegang

Wissenschaftlicher Mitarbeiter, Friedrich-Schiller-Universitat Je-na, bei Daniel Lenz, seit Oktober 2008.

Postdoctoral Fellow, Hebrew University Jerusalem, bei JonathanBreuer und Dan Mangoubi, Februar bis Juni 2011 und Oktober 2012bis September 2013.

1


Visiting Student Research Collaborator, Princeton University,bei Michael Aizenman und Simone Warzel, Oktober 2007 bis Mai2008.

Promotionsstipendiat, Stiftung der Deutschen Wirtschaft (SDW),Juli 2007 bis Juni 2010.

Wissenschaftlicher Mitarbeiter, Technische Universitat Chem-nitz, bei Daniel Lenz und Peter Stollmann, Juli 2006 bis Juni 2007.

Wissenschaftliche Hilfskraft, Fraunhofer Institut und TechnischeUniversitat Chemnitz, 2001 bis 2006.

Drittmittel

DFG-Projekt in Einzelforderung gemeinsam mit Daniel Lenz,Friedrich-Schiller-Universitat Jena, Geometrie diskreter Raume undSpektraltheorie nicht-lokaler Operatoren, (Sach- und Reisemittel, Dok-torandenstelle fur 3 Jahre, Post-Doktorandenstelle fur 1 Jahr) seitJuni 2012.

Stipendien

Golda Meir Fellowship, fur Forschungsaufenthalt an der HebrewUniversity Jerusalem, Oktober 2012 bis September 2013 (zusatzlichzu Stipendium der Hebrew University Grants von Jonathan Breuerund Dan Mangoubi).

Short visit grant der ESF fur Forschungsaufenthalt an der Tech-nischen Universitat Graz, Juli 2012.

Post-Doc Stipendium der Hebrew University, Jerusalem, Fe-bruar bis Juni 2011.

Promotionsstipendium, Stiftung der deutschen Wirtschaft (SDW),Juli 2007 bis Juni 2010.

Forschungsinteressen

Dirichletformen und Markov-Prozesse

Krummung und Spektrum von Graphen

Spektraltheorie von zufalligen Operatoren

2


Gutachtertatigkeit fur Zeitschriften

Analysis and Mathematical Physics

Annales Henri Poincare

Discrete & Computational Geometry

Discrete Mathematics

European Journal of Combinatorics

Journal of Combinatorial Theory, Series A

Journal of Functional Analysis

Linear Algebra and Its Applications

Mathematical Physics, Analysis and Geometry

Modern Physics Letters B

Nonlinear Analysis: Theory, Methods & Applications

Operators and Matrices

Physica A

Organisation von wissenschaftlichen Treffen

International Conference Fractal Geometry and Stochastics V, in Ta-barz, lokales Organisationskommittee.

Workshop Geometric aspects of probability and geometry, Friedrich-Schiller-Universitat Jena, September 2013.

Ein-Tages-Workshop Schrodinger operators, Friedrich-Schiller-UniversitatJena, Dezember 2011.

Doktorandensymposium innerhalb der Sommerschule Graphs and spec-tra, Technische Universitat Chemnitz, Juli 2011.

Doktorandensymposium innerhalb des Walkshop 2010, Friedrich-Schiller-Universitat Jena, September 2010.

Einladungen zu kurzeren Forschungsaufenthalten

Die Dauer der Aufenthalte betragt jeweils etwa 1 bis 2 Wochen.

Universite de Carthage, Bizerte Tunesia, Nabila Torki-Hamza, Marz2014.

University of Toronto, Balint Virag, Marz 2013.

Harvard University, Arbeitsgruppe Shing-Tung Yau, Marz 2013.

3


University of Connecticut, Arbeitsgruppe Alexander Teplayev, Febru-ar 2013.

Graduate Center CUNY, New York City, Jozef Dodziuk and RadoslawWojciechowski, Februar 2013.

Max-Planck-Institut Leipzig, Arbeitsgruppe Jurgen Jost, August 2012.

Technische Universitat Chemnitz, Arbeitsgruppen Peter Stollmannund Ivan Veselic, Juli 2012.

Technische Universitat Graz, Arbeitsgruppe Wolfgang Woess, Juli2012.

Universite Bordeaux 1, Sylvain Golenia, Mai 2012.

Max-Planck-Institut Leipzig, Arbeitsgruppe Jurgen Jost, Mai 2012.

Hebrew University Jerusalem, Jonathan Breuer und Dan Mangoubi,Dezember 2011.

Universitat Bielefeld, Arbeitsgruppe Alexander Grigor’yan, November2011.

Technische Universitat Munchen, Arbeitsgruppe Simone Warzel, Juli2011.

Universitat Bielefeld, Arbeitsgruppe Alexander Grigor’yan, Dezember2010.

Humboldt Universitat Berlin, Arbeitsgruppe Jochen Bruning, Dezem-ber 2010.

University of Lisbon, Group of Mathematical Physics Jean-ClaudeZambrini, November 2009.

Universitat Bielefeld, Arbeitsgruppe Michael Baake, Dezember 2008.

Technische Universitat Graz, Arbeitsgruppe Wolfgang Woess, Novem-ber 2008.

Graduate Center New York City University, Arbeitsgruppe Jozef Dod-ziuk, Mai 2008.

Technische Universitat Munchen, Simone Warzel, Oktober 2008.

Durham University, Norbert Peyerimhoff, September 2007.

Einladungen zu wissenschaftlichen Treffen mit Vortrag

Marz 2014, Konferenz in Tabarz Fractal Geometry and Stochastics V,’Cheeger’s inequality for unbounded graph Laplacians’.

Marz 2014, Universite de Carthage, Bizerte Tunesia, Cours pour Doc-torants, ’Lp Spectrum of Graphs’.

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Marz 2014, Universite de Carthage, Bizerte Tunesia, Journee-WorkShopGeometrie et Analyse sur les Graphes, ’Curvature and Spectrum onTessellating Graphs’.

Dezember 2013, Conference on Mathematical Technology of Networks- QGraphs 2013, ZiF Bielefeld, Intrinsic metrics on graphs.

November 2013, A colloquium on discrete curvature, C.I.R.M. Luminy,On the spectral theory of negatively curved planar graphs.

September 2013, Fall school Dirichlet forms, operator theory and ma-thematical physics Chemnitz, Minikurs Dirichlet forms on graphs.

Juli 2013, LMS Symposium, Graph Theory and Interactions, Durham,On negative curvature and spectrum of graph Laplacians.

September 2012, Workshop on Probability, Kansai University Osaka,Large time behavior of heat kernels.

September 2012, Conference Stochastic Analysis and Applications,Okayama, Essential spectra and volume growth of regular Dirichletforms.

August 2012, Conference Spectral Theory and Differential Operators,Graz, Volume growth and spectra of Dirichlet forms.

Januar 2012, Oberwolfach Mini-Workshop Boundary Value Problemsand Spectral Geometry, Curvature and spectrum on graphs.

Oktober 2011, Oberwolfach Workshop Correlations and Interactionsfor Random Quantum Systems, Absolutely continuous spectrum ontrees.

September 2011, BMS Summer School Random motion and randomgraphs, Berlin, Absolutely continuous spectrum for multi-type GaltonWatson trees.

September 2010, Walkshop and PhD Symposium, Jena, On the spec-tral theory of trees and tessellations.

September 2010, Conference QMath 11, Hradec Kralove, Absolutelycontinuous spectrum for substitution trees.

Juli 2010, Workshop Analysis on Graphs and its Applications, IsaacNewton Institute for Mathematical Sciences Cambridge, Absolutelycontinuous spectrum for trees of finite forward cone type.

Juni 2010, Workshop Random Schrodinger operators, Lausanne CIB,Stability of ac spectrum: Contraction properties of the recursion rela-tion.

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Mai 2010, AIMS Conference Dynamical Systems, Differential Equati-ons and Applications, Dresden, Stability of ac spectrum under randomperturbations on trees.

September 2009, Walkshop, Chemnitz, Dirichlet forms on discrete setsand absence of essential spectrum.

Juli 2009, Workshop Structure and Dynamics of Networks, Blaubeuren,Heat transport to the boundary on discrete graphs.

Juli 2009, Alp-Workshop, St. Kathrein, Random trees and absolutelycontinuous spectrum.

Juni 2009, Workshop Boundaries, Graz, Heat transport to the boun-dary.

November 2008, Workshop Structural Probability, Erwin-Schrodinger-Institut Wien, The Laplacian (plus potential) on trees and rapidlybranching graphs.

Einladungen zu Seminar- und Kolloquiumsvortragen

May 2014, Friedrich-Schiller-Universitat Jena, PhD Seminar, Intrinsicmetrics on graphs.

Mai 2014, Seminar Angewandte Analysis, Universitat Ulm University,Analysis and geometry on graphs.

Mai 2013, Friedrich-Schiller-Universitat Jena, PhD Seminar, Negativecurvature and spectrum of graphs.

Marz 2013, University of Toronto, Special lecture, Isoperimetric ine-qualities on graphs.

Marz 2013, University of Toronto, Toronto Probability Geometry Se-minar, Absolutely continuous spectrum of Galton-Watson trees.

Marz 2013, Harvard University, Differential Geometry Seminar, Int-rinsic metrics for Laplacians on graphs.

Februar 2013, University of Connecticut, Analysis and ProbabilitySeminar, Graphs of unbounded geometry and intrinsic metrics.

Februar 2013, Graduate Center CUNY, NY, Differential GeometrySeminar, Spectral consequences of upper curvature bounds on planargraphs.

Januar 2013, Technion Haifa, PDE and Applied Mathematics Semi-nar, Isoperimetric inequalities and volume growth estimates for un-bounded graph Laplacians.

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Dezember 2012, University of Haifa, Kolloqium, Periodic and randomSchrodinger operators on trees.

November 2012, Hebrew University Jerusalem, PDE Seminar, Cheegerinequalities and volume growth for unbounded graph Laplacians.

August 2012, Max-Planck-Institut Leipzig, Special Seminar, Geome-tric and spectral consequences of upper curvature bounds on planargraphs.

Juli 2012, Technische Universitat Chemnitz, Forschungsseminar Ana-lysis, Stochastik und Mathematische Physik, Anti-trees - the perfect(counter)example.

Juli 2012, Friedrich-Alexander-Universitat Erlangen, Negative curva-ture and discrete spectrum for graphs.

Juli 2012, Technische Universitat Graz, Seminar Mathematische Struk-turtheorie, Stability of spectral types for Galton Watson trees.

Mai 2012, Universite Bordeaux 1, Seminaire Analyse, On the spectraltheory of operators on trees.

Mai 2012, Max-Planck-Institut Leipzig, Special Seminar, Negativecurvature and spectrum of graph Laplacians.

Dezember 2011, Hebrew University Jerusalem, PDE Seminar, Abso-lutely continuous spectrum for Galton Watson trees.

November 2011, Technische Universitat Claustal, Oberseminar Ana-lysis und Spektraltheorie, Dirichlet forms on graphs.

Mai 2011, Technion Haifa, PDE and Applied Mathematics Seminar,On the long time behaviour of heat kernels.

April 2011, Weizmann Institute Rehovot, Geometric Functional Ana-lysis & Probability Seminar, Absolutely continuous spectrum for trees.

Marz 2011, Hebrew University Jerusalem, PDE Seminar, Curvatureand spectrum for graphs.

Dezember 2010, Friedrich-Alexander-Universitat Erlangen, Absolutelycontinuous spectrum for trees.

Dezember 2010, Universitat Bielefeld, Oberseminar Geometric Ana-lysis, Curvature and spectrum for graphs.

November 2010, Friedrich-Schiller-Universitat Jena, Seminar Analy-sis, Geometrie und Stochastik, Planar graphs of non-positive curva-ture.

Juni 2010, Humboldt Universitat Berlin, Seminar Geometrische Ana-lysis und Spektraltheorie, Cheeger constants, exponential growth andspectrum of planar graphs.

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Juni 2010, Technische Universitat Chemnitz, Forschungsseminar Ana-lysis, Stochastik und Mathematische Physik, Cheeger constants, ex-ponential growth and spectrum of planar graphs.

Mai 2010, Friedrich-Schiller-Universitat Jena, Seminar Analysis, Geo-metrie und Stochastik, Discrete Spectrum for Schrodinger operatorson graphs.

Januar 2010, Friedrich-Schiller-Universitat Jena, Seminar Analysis,Cheegerkonstanten, exponentielles Wachstum und Spektrum von pla-naren Graphen.

November 2009, University of Lisbon, Seminar of Mathematical Phy-sics, Cheeger constants, exponential growth and spectrum of planargraphs.

Oktober 2009, Hebrew University Jerusalem, PDE Seminar, Trees offinite vertex type and absolutely continuous spectrum.

Dezember 2008, Universitat Bielefeld, Seminar Mathematik in denNaturwissenschaften, The Laplacian on rapidly branching graphs.

November 2008, Technische Universitat Graz, Seminar Mathemati-sche Strukturtheorie, The Laplacian on rapidly branching graphs.

November 2008, Universitat Wien, Seminar Analysis, The Laplacianon rapidly branching graphs.

Oktober 2008, Ludwig-Maximilian-Universitat Munchen, Seminar Ana-lysis und Zufall, The Laplacian on rapidly branching graphs.

April 2008, Houston Rice University, Geometry-Analysis Seminar,The Laplacian on rapidly branching graphs.

Februar 2008, Graduate Center CUNY, NY, Differential GeometrySeminar, The Laplacian on rapidly branching graphs.

September 2007, Durham University, Geometry Seminar, Spectral pro-perties of rapidly branching graphs.

Betreuung und Mitbetreuung von Abschlussarbeiten

Florentin Munch, Li-Yau inequalities on graphs, Master 2014.

Melchior Wirth, Does diffusion determine the graph structure?, Ba-chelor 2013.

Oliver Siebert, Spectra of lamplighter random walks associated withpercolation, Bachelor 2013.

Ricardo Kalke, Ricci curvature on graphs, Staatsexamen 2012.

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Florentin Munch, Ultrametrische Cantormengen und Rander von Bau-men, Bachelor 2012.

Marcel Schmidt, Themen im Bereich Globale Eigenschaften von Di-richletformen, Dissertation.

Lehre

SS 2014 Ubung Analysis II fur Mathematiker und Physiker.

WS 2013/14 Vorlesung Anwendung von Operatortheorie, zweistundig.

WS 2013/14 Ubung Analysis I fur Mathematiker.

SS 2012 Vorlesung Anwendung von Operatortheorie, vierstundig, (uber-durchschnittlich positive Lehrevaluation).

WS 2011/12 Ubung Analysis III fur Mathematiker, Ubung AnalysisIII fur Physiker (uberdurchschnittlich positive Lehrevaluation).

WS 2010/11 Ubung Analysis I fur Mathematiker (uberdurchschnitt-lich positive Lehrevaluation).

WS 2009/10 Ubung Analysis III fur Physiker.

SS 2009 Ubung Analysis II fur Physiker, Seminar Zufallige Schrodin-ger Operatoren.

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Schriftenverzeichnis

371

Schriftenverzeichnis - Matthias Keller

Publikationen in referierten Journalen

1. Diffusion determines the recurrent graph (mit Daniel Lenz, Marcel Schmidt,Melchior Wirth), erscheint in Advances in Mathematics.

2. An invitation to trees of finite cone type: random and deterministic opera-tors, (mit Daniel Lenz und Simone Warzel), erscheint in Markov Processesand Related fields.

3. Harmonic functions of general graph Laplacians (mit Bobo Hua), erscheintin Calculus of Variations and Partial Differential Equations.

4. On the lp spectrum of Laplacians on graphs (mit Frank Bauer, Bobo Hua),Advances in Mathematics, 248, Issue 25 (2013), 717-735.

5. Note on basic features of large time behaviour of heat kernels (mit DanielLenz, Hendrik Vogt, Rados law Wojciechowski), erscheint in Journal fur diereine und angewandte Mathematik (Crelle’s Journal).

6. Cheeger inequalities for unbounded graph Laplacians (mit Frank Bauer, Ra-dos law Wojciechowski), erscheint in Journal of the European MathematicalSociety.

7. Volume growth and bounds for the essential spectrum for Dirichlet forms(mit Sebastian Haeseler, Rados law Wojciechowski), Journal of the LondonMathematical Society, Volume 88, Issue 3 (2013), 883–898.

8. Spectral analysis of certain spherically homogeneous graphs (mit JonathanBreuer), Operators and Matrices, Volume 7, Number 4 (2013), 825–847.

9. A note on self-adjoint extensions of the Laplacian on weighted graphs(mit Xueping Huang, Jun Masamune, Radoslaw Wojciechowski), Journalof Functional Analysis, Volume 265, Issue 8 (2013), 1556-1578.

10. Volume growth, spectrum and stochastic completeness of infinite graphs(mit Daniel Lenz, Rados law Wojciechowski), Mathematische Zeitschrift,Volume 274, Issue 3 (2013), 905-932.

11. Laplacians on infinite graphs: Dirichlet and Neumann boundary conditions(mit Sebastian Haeseler, Daniel Lenz, Rados law Wojciechowski), Journalof Spectral Theory, Volume 2, Issue 4, (2012) 397-432.

12. Absolutely continuous spectrum for multi-type Galton Watson trees, An-nales Henri Poincare, Volume 13, Issue 8 (2012), 1745-1766.

13. Absolutely continuous spectrum for random operators on trees of finite conetype (mit Daniel Lenz and Simone Warzel), Journal d’Analyse MathematiqueVolume 118, Issue 1, (2012) 363-396.

14. On the spectral theory of trees with finite cone type (mit Daniel Lenzand Simone Warzel), Israel Journal of Mathematics, Volume 194, Issue 1,(2013), 107–135.

15. Dirichlet forms and stochastic completeness of graphs and subgraphs (mitDaniel Lenz), Journal fur die reine und angewandte Mathematik (Crelle’sJournal), Volume 2012, Issue 666, 189-223.

16. Curvature, geometry and spectral properties of planar graphs, Discrete &Computational Geometry, Volume 46, Issue 3 (2011), 500-525.

17. Generalized solutions and spectrum for Dirichlet forms on graphs (mit Seba-stian Haeseler), Random Walks, Boundaries and Spectra, Progress in Pro-bability (2011), Birkhauser, 181-201.

18. Cheeger constants, growth and spectrum of locally tessellating planar graphs(mit Norbert Peyerimhoff), Mathematische Zeitschrift, Volume 268, Issue3-4 (2011), 871-886.

19. Unbounded Laplacians on Graphs: Basic Spectral Properties and the HeatEquation (mit Daniel Lenz), Mathematical modeling of natural phenomena:Spectral Problems, Volume 5, No. 4 (2010), 198-224.

20. The essential spectrum of the Laplacian on rapidly branching tessellations,Mathematische Annalen, Volume 346, Issue 1 (2010), 51-66.

Vorveroffentlichungen

21. Intrinsic metrics on graphs - A survey.

22. A overview of curvature bounds and spectral theory of planar tessellations.

23. Eigenvalue asymptotics for Schrodinger operators on sparse graphs (mitMichel Bonnefont, Sylvain Golenia), arXiv:1311.7221.

24. Graphs of finite measure (mit Agelos Georgakopoulos, Sebastian Haeseler,Daniel Lenz, Radoslaw Wojciechowski), arXiv:1309.3501.

25. A Feynman-Kac-Ito formula for magnetic Schrodinger operators on graphs(mit Batu Guneysu, Marcel Schmidt), arXiv:1301.1304.

Abschlussarbeiten und sonstige Schriften

26. On the spectral theory of operators on trees, Dissertation 2010, Friedrich-Schiller-Universtitat Jena.

27. Produkte zufalliger Matrizen und der Lyapunov-Exponent, Diplomarbeit2006, Technische Universitat Chemnitz.

28. Absolutely continuous spectrum on trees – random potentials, random hop-ping and Galton-Watson trees, Oberwolfach Report No. 50/2011, Oktober2011.

29. Curvature and spectrum on graphs, Oberwolfach Report No. 02/2012, Ja-nuar 2012.

Ehrenwortliche Erklarung

Ich bestatige, dass mir die geltende Habilitationsordnung der Fa-kultat bekannt ist. Ich versichere, dass ich die eingereichte Habili-tation selbstandig verfasst, dass ich die Habilitation selbst angefer-tigt habe, keine Textabschnitte oder Ergebnisse eines Dritten odereigener Prufungsarbeiten ohne Kennzeichnung ubernommen und allebenutzten Hilfsmittel, personlichen Mitteilungen und Quellen in derArbeit angegeben habe. Weiterhin erklare ich, dass die Hilfe eines Ha-bilitationsberaters nicht in Anspruch genommen wurde und dass Dritteweder unmittelbar noch mittelbar geldwerte Leistungen von mir furArbeiten erhalten haben, die im Zusammenhang mit dem Inhalt dervorgelegten Habilitation stehen. Außerdem bestatige ich, dass ich dieHabilitation noch nicht als Prufungsarbeit fur eine staatliche oder an-dere wissenschaftliche Prufung eingereicht habe. Weiterhin versichereich, dass nicht an anderer Stelle ein Habilitationsverfahren fur das glei-che Fachgebiet beantragt worden oder erfolglos beendet worden ist.

Jena, den 18.08.2014

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Documents

On the Geometry and Analysis of Graphs...On the Geometry and Analysis of Graphs Habilitationsschrift vorgelegt am 19. 08. 2014 der Fakult at fur Mathematik und Informatik der Friedrich-Schiller-Universit