UNIVERSITY LECTURE SERIES VOLUME 71

Introduction to Analysis on Graphs

Alexander Grigor’yan

Introduction to Analysis on Graphs

10.1090/ulect/071

Introduction to Analysis on Graphs

Alexander Grigor’yan

UNIVERSITY LECTURE SERIES VOLUME 71

EDITORIAL COMMITTEE

Jordan S. EllenbergWilliam P. Minicozzi II (Chair)

Robert GuralnickTatiana Toro

2010 Mathematics Subject Classification. Primary 05C50, 05C63, 05C76, 05C81, 60J10.

For additional information and updates on this book, visitwww.ams.org/bookpages/ulect-71

Library of Congress Cataloging-in-Publication Data

Names: Grigoryan, A. (Alexander), author.Title: Introduction to analysis on graphs/Alexander Grigor’yan.Description: Providence, Rhode Island : American Mathematical Society, [2018] | Series: Univer-

sity lecture series ; volume 71 | Includes bibliographical references and index.Identifiers: LCCN 2018001105 | ISBN 9781470443979 (alk. paper)Subjects: LCSH: Graph theory. | Laplace transformation. | Finite groups. | AMS: Combinatorics

– Graph theory – Graphs and linear algebra (matrices, eigenvalues, etc.). msc | Combinatorics– Graph theory– Infinite graphs. msc | Combinatorics – Graph theory – Graph operations (linegraphs, products, etc.). msc | Combinatorics – Graph theory – Random walks on graphs. msc |Probability theory and stochastic processes – Markov processes –Markov chains (discrete-timeMarkov processes on discrete state spaces). msc

Classification: LCC QA166.G7485 2018 | DDC 511/.5–dc23LC record available at https://lccn.loc.gov/2018001105

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10 9 8 7 6 5 4 3 2 1 23 22 21 20 19 18

Contents

Preface vii

Chapter 1. The Laplace operator on graphs 11.1. The notion of a graph 11.2. Cayley graphs 51.3. Random walks 81.4. The Laplace operator 191.5. The Dirichlet problem 22

Chapter 2. Spectral properties of the Laplace operator 272.1. Green’s formula 272.2. Eigenvalues of the Laplace operator 282.3. Convergence to equilibrium 342.4. More about the eigenvalues 392.5. Convergence to equilibrium for bipartite graphs 422.6. Eigenvalues of Zm 432.7. Products of graphs 452.8. Eigenvalues and mixing time in Z

nm, m odd. 49

2.9. Eigenvalues and mixing time in a binary cube 51

Chapter 3. Geometric bounds for the eigenvalues 533.1. Cheeger’s inequality 533.2. Eigenvalues on a path graph 583.3. Estimating λ1 via diameter 613.4. Expansion rate 63

Chapter 4. Eigenvalues on infinite graphs 734.1. Dirichlet Laplace operator 734.2. Cheeger’s inequality 764.3. Isoperimetric and Faber-Krahn inequalities 784.4. Estimating λ1 (Ω) via inradius 794.5. Isoperimetric inequalities on Cayley graphs 824.6. Solving the Dirichlet problem by iterations 86

Chapter 5. Estimates of the heat kernel 895.1. The notion and basic properties of the heat kernel 895.2. One-dimensional simple random walk 915.3. Carne-Varopoulos estimate 965.4. On-diagonal upper estimates of the heat kernel 995.5. On-diagonal lower bound via the Dirichlet eigenvalues 1075.6. On-diagonal lower bound via volume growth 112

v

vi CONTENTS

5.7. Escape rate of random walk 114

Chapter 6. The type problem 1176.1. Recurrence and transience 1176.2. Recurrence and transience on Cayley graphs 1226.3. Volume tests for recurrence 1236.4. Isoperimetric tests for transience 128

Chapter 7. Exercises 131

Bibliography 143

Index 149

Preface

This book is based on a semester lecture course Analysis on Graphs that Itaught a number of years ago at the Department of Mathematics of the Universityof Bielefeld. The purpose of the book is to provide an introduction to the subjectof the discrete Laplace operator on locally finite graphs. It should be accessible toundergraduate and graduate students with enough background in linear algebra,analysis and elementary probability theory.

The book starts with elementary material at the level of first semester mathe-matics students, and concludes with the results proved in the mathematical litera-ture in 1990s. However, the book covers only some selected topics about the discreteLaplacian and is complementary to many existing books on similar subjects.

Let us briefly describe the contents of the book.In Chapter 1 we give the definition and prove some basic properties of the dis-

crete Laplace operator such as solvability of the Dirichlet problem and the existenceof the associated random walk (= a reversible Markov chain).

In Chapter 2 we are concerned with the eigenvalues of the Laplace operator onfinite graphs and their relation to the rate of convergence to the equilibrium of thecorresponding random walk.

Chapter 3 contains some estimates of the eigenvalues on finite graphs, in par-ticular, Cheeger’s inequality, as well as the relation of eigenvalues to the expansionrate of subsets of graphs [37], [39].

In Chapter 4 we deal with the Laplace operator on infinite graphs and itsrestriction to finite domains – the Dirichlet Laplacian. The central topic is therelation between the eigenvalues of the Dirichlet Laplacian and the isoperimetricproperties of the graph, which is based on a version of Cheeger’s inequality. InSection 4.5 we prove a beautiful theorem of Coulhon and Saloff-Coste [51] aboutisoperimetric inequalities on Cayley graphs.

Chapter 5 is devoted to heat kernel estimates on infinite graphs, where the heatkernel is the density of the transition probability of the random walk with respect tothe underlying measure (for example, the degree measure). In the case of a simplerandom walk in Z we obtain the estimates directly from the definition by meansof Stirling’s formula. In Section 5.3 we prove a universal Gaussian upper bound ofthe heat kernel that is due to Carne [32] and Varopoulos [136] . In Section 5.4 weprove the on-diagonal upper bound of the heat kernel of [47], [81] assuming thatthe graph satisfies a Faber-Krahn inequality.

In Sections 5.5-5.6 we prove some lower bounds of the heat kernel of [49], [113].In Section 5.7 we use the heat kernel techniques to prove a universal upper bound forescape rate of the random walk on graphs with the polynomial volume growth. Thiscan be regarded as a far-reaching generalization of the Hardy-Littlewood

√n logn-

estimate for the escape rate of a simple random walk in Zm, which was obtained in

vii

viii PREFACE

1914 (ten years before Khinchin’s law of the iterated logarithm). For graphs withpolynomial volume growth the

√n log n-estimate is sharp as was shown in [15].

In Chapter 6 we are concerned with the problem of deciding whether the ran-dom walk is recurrent or transient. Here we give a number of analytic conditions forrecurrence and transience. In particular, the heat kernel bounds from the previouschapter lead immediately to the celebrated theorem of Polya: the simple randomwalk in Z

m is recurrent if and only ifm ≤ 2 (Section 6.1). A far-reaching generaliza-tion of Polya’s theorem is the Varopoulos criterion [135] for recurrence on Cayleygraphs that is presented in Section 6.2. In the remaining part of this chapter, weprove for general graphs Nash-Williams’ [115] and volume tests for recurrence (thelatter being a discrete version of a theorem of Cheng-Yau [33] about parabolicityof Riemannian manifolds), as well as an isoperimetric test for transience [72].

Chapter 7 contains exercises that were actually used for homework in the afore-mentioned lecture course. Solutions to all exercises are available on my home page.

Some remarks are due concerning the bibliography. Initially I planned to limitmyself to a minimal bibliography list containing only necessary references from thetext. However, it was suggested by an anonymous referee that the bibliographyshould contain also references to the sources in adjacent areas thus providing abroader coverage of topics of analysis on graphs. Furthermore, the referee hadkindly offered a long list of such references, which greatly facilitated my work onthe bibliography list. Hence, here is a list of sources for further reading.

• Classical (combinatorial) graph theory: [31], [38], [59], [122], [123].• Various aspects of analysis on graphs: [29], [45], [46], [52], [53], [64],[65], [66], [129], [130].

• Spectral theory on graphs: [6], [19], [21], [22], [24], [27], [28], [30], [35],[39], [40], [42], [43], [44], [45], [52], [53], [67], [70], [90], [92], [93], [105],[106], [114], [125], [132], [133], [134].

• Potential-theoretic aspects of graphs: [63], [96], [97], [127], [137], [139].• Analysis on Cayley and Schreier graphs: [16], [17], [49], [51], [54], [65],[66], [71], [83], [84], [85], [112], [118], [121], [119].

• Random processes on graphs: [9], [10], [15], [86], [120], [124], [131],[139], [140].

• Heat kernels on graphs: [10], [11], [12], [13], [14], [47], [48], [50], [55],[56], [68], [69], [81], [82], [87], [88], [100], [101], [102], [116], [126]

• Curvature on graphs: [20], [23], [41], [94], [95], [107], [108], [109].• Homology theory on graphs: [4], [7], [60], [75], [76], [77], [78], [79], [80].• Analysis on metric/quantum graphs: [26], [27], [64], [67], [98].• Analysis on fractals and ultra-metric spaces: [3], [8], [25], [64], [73], [99],[128].

Alexander Grigor’yan, March 2018

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Index

backward equation, 16

binary cube, 8, 32boundary condition, 23

Carne-Varopoulos estimate, 96Chebyshev polynomials, 97

Cheegerconstant on finite graphs, 53

constant on infinite graphs, 76

inequality on finite graphs, 54inequality on infinite graphs, 77

co-area formula, 54complement Ωc, 22

convergence rate, 88

degree, 1

diameter, 61Dirichlet Laplace operator LΩ, 73

Dirichlet problem, 22, 86double counting, 1

edge, 1edge boundary, 53

edge generating set, 6eigenfunction, 29

eigenvalues, 28

of a binary cube, 51of the Dirichlet Laplace operator, 75

of the Laplace operator, 32of the Markov operator, 75

of Zm, 44of Zn

m, 49

on products, 47, 48

equilibrium measure, 18, 38escape rate, 114

expansion rate, 65

Faber-Krahn inequality, 78, 80

on Cayley graphs, 83on Zm, 83

forward equation, 16

graph, 1

bipartite, 31, 32, 42Cayley, 6

complete, 39

complete bipartite Kn,m, 2

complete Kn, 2

connected, 4

cycle Cq, 7

finite, 1

infinite, 73

locally finite, 1

path graph, 58

regular, 8

simple, 1

weighted, 3

Zn, 2

graph distance, 4

Green’s formula

for finite graphs, 27

for infinite graphs, 74

group, 5

Zn, 6

Zq, 5

heat kernel, 89

lower bound for sup pn (x, x), 107

on-diagonal lower bound

on Cayley graphs, 107

on polycyclic groups , 111

on Vicsek tree, 110

on Zm, 109

via volume, 112

on-diagonal upper bound, 100

on Cayley graphs, 106

on Vicsek tree, 100

on Zm, 100

inner product, 31

inradius, 79

isoperimetric inequality, 78

on Cayley graphs, 82

on Zm, 83

Laplace operator

on a graph, 20

positive definite, 31

weighted, 20

149

150 INDEX

Markov chain, 10Markov kernel, 9, 20

on products, 46reversible, 14

Markov operator, 21, 34Markov property, 10maximum principle, 23

strong, 119minimum principle, 23mixing time, 18, 38

on a binary cube, 52on Zn

m, 50

path, 4Polya’s theorem, 19, 117product

of graphs, 45of regular graphs, 47weighted, 46

product of groups, 5

random walkon Z, 9, 91recurrent, 19, 117simple, 9transient, 19, 117

rate of convergence, 35Rayleigh quotient, 30recurrence

Nash-Williams test, 123on Cayley graphs, 122volume test, 126

residue, 5

spectral radius, 36, 63spectrum, 28subharmonic, 23superharmonic, 23

trace, 40transience

isoperimetric test, 128on Cayley graphs, 122

transition function, 13, 35type problem, 117

vertex, 1Vicsek tree, 81

weight

of a vertex, 15of edges, 3of vertices, 3simple, 3

Anybody who has ever read a mathematical text of the author would agree that his way of

presenting complex material is nothing short of marvelous. This new book showcases again the

author’s unique ability of presenting challenging topics in a clear and accessible manner, and of

guiding the reader with ease to a deep understanding of the subject.

—Matthias Keller, University of Potsdam

A central object of this book is the discrete Laplace operator on finite and infinite graphs.

The eigenvalues of the discrete Laplace operator have long been used in graph theory

as a convenient tool for understanding the structure of complex graphs. They can also

be used in order to estimate the rate of convergence to equilibrium of a random walk

(Markov chain) on finite graphs. For infinite graphs, a study of the heat kernel allows to

solve the type problem—a problem of deciding whether the random walk is recurrent or

transient.

This book starts with elementary properties of the eigenvalues on finite graphs,

continues with their estimates and applications, and concludes with heat kernel esti-

mates on infinite graphs and their application to the type problem.

The book is suitable for beginners in the subject and accessible to undergraduate and

graduate students with a background in linear algebra I and analysis I. It is based on a

lecture course taught by the author and includes a wide variety of exercises. The book

will help the reader to reach a level of understanding sufficient to start pursuing research

in this exciting area.

For additional information

and updates on this book, visit

www.ams.org/bookpages/ulect-71

ULECT/71