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Distance Spectra of Graphs

Leslie Hogben

Iowa State University and American Institute of Mathematics

47th Southeastern International Conference onCombinatorics, Graph Theory & Computing

Florida Atlantic UniversityMarch 8, 2016

Joint work with G. Aalipour, A. Abiad, Z. Berikkyzy, J. Cummings, J.De Silva, W. Gao, K. Heysse, F.H.J. Kenter, J.C.-H. Lin, and M. Tait

at the Graduate Research Workshop in Combinatorics (GRWC)

Leslie Hogben (Iowa State University and American Institute of Mathematics) 1 of 53

I. Introduction(A) Data communication problem: Loop switching(B) Distance matrices(C) Distance spectra results

II. Unimodality of distance characteristic polynomial coefficients for trees(A) Unimodality and Peak Conjectures of Graham and Lovász(B) Collins’ counterexample to the Peak Conjecture(C) Proof of the Graham/Lovász Unimodality Conjecture(D) Collins/Shor Peak Conjecture(E) Graphs that are not trees

III. On distance spectra(A) Optimistic strongly regular graphs(B) Distance regular graphs having one positive distance eigenvalue(C) Distance determinants and inertias of generalized barbells andlollipops(D) Number of distinct distance eigenvalues


Section I: Introduction

(A) Data communication problem: Loop switching

(B) Distance matrices

(C) Distance spectra results


(A) Loop switching

[Graham, Pollak; 1971]

Problem: Find the best routing for messages through acommunication network.

Model:

I Subscribers or computer terminals are on one-way loops (localloops) that are connected to each other and to regional loops,which are connected to each other and to a national loop.

I A message from a subscriber on one loop is usually destinedfor a subscriber on another loop.

I To move through the system, a message proceeds around itsloop until it reaches a switching point, at which point itchooses to continue on its loop or switch to another loop.

Question: Find a simple efficient routing strategy for loopswitching.


Graham and Pollak’s solution for loop switching

I Model the loop system with a graph.

I Use the distance matrix of the graph to find addresses.

I Use the addresses to chose at each junction whether or not toswitch.

I Routing method is simple to apply.

I Message takes shortest path between loops in same region.

I Applies to any loop configuration.


Graph interpretation of loop system

R.L. Graham and H.O. Pollak. On the addressing problem for loopswitching. Bell Syst. Tech. J. 50 (1971), 2495–2519.http://www.math.ucsd.edu/∼ronspubs/71 05 loop switching.pdf


http://www.math.ucsd.edu/~ronspubs/71_05_loop_switching.pdf

(B) Distance matrices

All graphs are simple, undirected, finite, and connected.G = (V ,E ) is a graph of order n with V = {v1, . . . , vn}.I The distance d(vi , vj) between vertices vi and vj is the length

of a shortest path between vi and vj .

I D(G ) = [d(vi , vj)] is the distance matrix of G (n × n matrix).

G1

2

3 4 5 D(G ) =

0 1 1 2 31 0 1 2 31 1 0 1 22 2 1 0 13 3 2 1 0


Matrices, eigenvalues, spectra

Let A be a real symmetric matrix.

I Real number λ is an eigenvalue of A if Aw = λw for anonzero vector w.

I pA(x) = det(xI − A) is the characteristic polynomial of G .I λ is an eigenvalue of matrix A if and only if pA(λ) = 0.

I The multiplicity multA(λ) is the exponent of (x − λ) in pA(x).I Because A is real and symmetric, multA(λ) = null(A− λI ).I The multiset of eigenvalues, appearing with multiplicity, is the

spectrum, spec(A), of A.

I The inertia of A is the triple (n+, n0, n−), with the entriesindicating the number of positive, zero, and negativeeigenvalues (counting multiplicities).


Distance spectra: definitions and examples

I pD(G)(x) = det(xI −D(G )) is the distance characteristicpolynomial of G .

I A distance eigenvalue of G is an eigenvalue of D(G ).I specD(G ) := spec(D(G )) is the distance spectrum of G (the

multiset of distance eigenvalues, appearing with multiplicity).

G1

2

3 4 5 D(G ) =

0 1 1 2 31 0 1 2 31 1 0 1 22 2 1 0 13 3 2 1 0

specD(G ) = {−4.26261,−1.17742,−1,−0.568579, 7.00861}

Distance Spectra SurveyM. Aouchiche and P. Hansen. Distance spectra of graphs: Asurvey. Linear Algebra Appl., 458 (2014), 301–386.


(C) Distance spectra results: Loop switching

Theorem (Graham, Pollak; 1971)

Given any system of n loops with maximum distance s betweenloops, there is a system of addresses of length no more thans(n − 1) with the property that a minimal path between loops canbe obtained by switching to an adjacent loop if and only if theHamming distance to the destination is decreased by one.


If the graph of the loop system is one of the following graphs, thenaddresses of length |V (G )| − 1 suffice:I Tree

I Complete graph


Distance spectra of trees

The proof that loop system addresses of length n − 1 suffice whenthe graph is a tree (of order n) relies on the graph having onepositive (and no zero) distance eigenvalues.


For a tree T of order n ≥ 2,

detD(T ) = (−1)n−1(n − 1)2n−2,

and the inertia of D(T ) is (n+, n0, n−) = (1, 0, n − 1).

Generalization to graphs:

I The determinant of the distance matrix depends only on theblocks of the graph.

I Conjectured in [Hosoya, Murakami, Gotoh; 1973].

I Proved in [Graham, Hoffman, Hosoya; 1977].Leslie Hogben (Iowa State University and American Institute of Mathematics) 11 of 53

Distance spectra of trees

I The distance determinant is a coefficient of the distancecharacteristic polynomial.

I Signs of the coefficients of the distance characteristicpolynomial of a tree determined in [Edelberg, Garey, Graham;1976]

I Coefficients of the distance characteristic polynomial of a treecomputed in terms of subforests in [Graham, Lovász; 1978]

Theorem (Merris; 1990)

For T a tree, L(T ) the line graph of T , A(L(T )) the adjacencymatrix of L(T ), and K := 2I +A(L(T )). The eigenvalues of−2K−1 interlace the eigenvalues of D(T ) (with D(T ) having onemore).


Distance spectra of cartesian products

I G = (V ,E ) and G ′ = (V ′,E ′): G �G ′ has vertex set V ×V ′,and vertices (u, u′) and (v , v ′) are adjacent if (u = v and{u′, v ′} ∈ E ′) or (u′ = v ′ and {u, v} ∈ E ).

I G is transmission regular if D(G )1 = ρ1 (where ρ is theconstant row sum of D(G ) and 1 is the all ones vector).

Theorem (Indulal; 2009)

Suppose G and G ′ are transmission regular graphs withspecD(G ) = {ρ, θ2, . . . , θn} and specD(G ′) = {ρ′, θ′2, . . . , θ′n′}.Then

specD(G �G′) = {n′ρ+ nρ′} ∪ {n′θ2, . . . , n′θn} ∪

{nθ′2, . . . , nθ′n′} ∪ {0((n−1)(n′−1))}.


Section II: Unimodality of the distance characteristicpolynomial coefficients for trees

(A) Unimodality and Peak Conjectures of Graham and Lovász

(B) Collins’ counterexample to the Peak Conjecture

(C) Proof of the Graham/Lovász Unimodality Conjecture

(D) Collins/Shor Peak Conjecture

(E) Graphs that are not trees


(A) Unimodality and Peak Conjectures:Definitions

G is a graph of order n.

I Graham and Lovász used the polynomial

det(D(G )−xI ) =: (−1)nxn+δn−1(G )xn−1+· · ·+δ1(G )x+δ0(G )

I pD(G)(x) = det(xI −D(G )) =xn + (−1)nδn−1(G )xn−1 + · · ·+ (−1)nδ1(G )x + (−1)nδ0(G )

I For 0 ≤ k ≤ n − 2, the normalized coefficients are

dk(G ) :=

(1

2n−2

)2k |δk(G )|.

I [Graham, Lovász; 1978] If T is a tree, then the normalizedcoefficients represent counts of certain subforests of the tree.

I [Graham, Lovász; 1978] If T is a tree, thendk(T ) = (−1)n−1δk(T )/2n−k−2


Graham and Lovász’ Unimodality and Peak Conjectures

The real sequence a0, a1, a2, . . . , an is unimodal if there is a k suchthat ai−1 ≤ ai for i ≤ k and ai ≥ ai+1 for i ≥ k .

Conjecture (Graham, Lovász; 1978)

For a tree T of order n ≥ 3:I Unimodality Conjecture The sequence of normalized

coefficients d0(T ), . . . , dn−2(T ) is unimodal.

I Peak Conjecture The peak of the unimodal sequenced0(T ), . . . , dn−2(T ) occurs at

⌊n2

⌋.


(B) Collins’ counterexample to the Peak Conjecture

Theorem (Collins; 1989)

I For both stars and paths the sequence d0(T ), . . . , dn−2(T ) isunimodal.

I For stars the peak is at⌊n2

⌋.

I For paths the peak is at approximately(1− 1√

5

)n ≈ 0.553n


(C) Proof of the Unimodality Conjecture:Key definitions and results

I a0, a1, a2, . . . , an ∈ R is unimodal if there is a k such thatai−1 ≤ ai for i ≤ k and ai ≥ ai+1 for i ≥ k.

I a0, a1, a2, . . . , an ∈ R is log-concave if a2j ≥ aj−1aj+1 for allj = 1, . . . , n − 1.

I a0, a1, a2, . . . , an is the coefficent sequence of polynomialp(x) = a0 + a1x + · · ·+ an−1xn−1 + anxn.

The following results appears in various surveys about unimodality.

Theorem

(i) If p(x) is a real-rooted polynomial, then the coefficientsequence of p(x) is log-concave.

(ii) If a0, a1, a2, . . . , an is positive and log-concave, thena0, a1, a2, . . . , an is unimodal.


Unimodality theorem

Theorem (Aalipour, Abiad, Berikkyzy, Hogben, Kenter, Lin, Tait)

Let T be a tree of order n ≥ 3.(i) The coefficient sequence of the distance characteristic

polynomial pD(T )(x) is log-concave.

(ii) The sequence |δ0(T )|, . . . , |δn−2(T )| of absolute values ofcoefficients of the distance characteristic polynomial islog-concave and unimodal.

(iii) The sequence d0(T ), . . . , dn−2(T ) of normalized coefficientsof the distance characteristic polynomial is log-concave andunimodal.


Proof of the Unimodality Theorem: Distance coefficients

I pD(T )(x) is real-rooted

I The coefficient sequence (−1)nδ0(T ), . . . , (−1)nδn−2(T ), 0, 1is log-concave.

I (−1)nδ0(T ), . . . , (−1)nδn−2(T ) is log-concave.I (−1)n−1δk(T ) > 0 for 0 ≤ k ≤ n − 2.I (−1)nδk(T ) < 0 for 0 ≤ k ≤ n − 2.I All of the terms (−1)nδ0(T ), . . . , (−1)nδn−2(T ) are negative.I {|δk(T )|}n−2k=0 is log-concave and positive.I |δ0(T )|, . . . , |δn−2(T )| is unimodal.


Proof: Normalized distance coefficients

Observation

Let a0, a1, a2, . . . , an be a sequence of real numbers, let c and s benonzero real numbers, and define bk = sc

kak . Thena0, a1, a2, . . . , an is log-concave if and only if b0, b1, b2, . . . , bn islog-concave.

Proof continued

I |δ0(T )|, . . . , |δn−2(T )| is log-concave.I dk(T ) =

(1

2n−2

)2k |δk(T )| for k = 0, . . . , n − 2.

I d0(T ), . . . , dn−2(T ) is log-concave and positive.

I d0(T ), . . . , dn−2(T ) is unimodal.


(D) Collins/Shor Peak Conjecture

Conjecture (Collins; 1989; attributed to Shor)

For every tree T of order at least 3, the location of the peak of theunimodal sequence of normalized distance coefficients

d0(T ), . . . , dn−2(T )

is between ⌊n2

⌋and

⌈n − n√

5

⌉.

Computations on Sage confirm this conjecture for all trees of orderat most 20.


(E) Graphs that are not trees

2 4 6 8 10 12

5000

10 000

15 000

The Heawood graph and a plot of its normalized distancecoefficients,

81, 924, 3794, 5460, 3801, 14728, 1848,

17928, 6545, 9492, 9114, 3164, 441.


Section III: On distance spectra

(A) Optimistic strongly regular graphs

(B) Distance regular graphs having one positive distanceeigenvalue

(C) Distance determinants and inertias of barbells and lollipops

(D) Number of distinct distance eigenvalues


(A) Optimistic strongly regular graphs

I G is optimistic if G has more positive than negative distanceeigenvalues.

I [Graham, Lovász; 1978] asked whether optimistic graphsexist.

I [Azarija; 2014] exhibited families of strongly regularoptimistic graphs.

I [AABCDGHHKLT, 2016] characterized which strongly regulargraphs are optimistic in terms of their parameters.


Strongly regular graphs

I G is an (n, k, λ, µ) strongly regular graph (SRG) ifI G has n vertices,I G Is k-regular,I every two adjacent vertices have λ common neighbors, andI every two nonadjacent vertices have µ common neighbors.

I At most 3 adjacency eigenvalues of a strongly regular graph:I ρ = k with multiplicity 1.I θ > 0 with multiplicity mθ ≥ 0 (mθ = 0 iff Kn).I τ < 0 with multiplicity mτ > 0.

I Formulas for θ, τ,mθ,mτ in terms of parameters (n, k, λ, µ)are well known.


Distance matrices of strongly regular graphs

Suppose G is a connected (n, k, λ, µ)-SRG, so µ ≥ 1 (or G = Kn).I The maximum distance between vertices is 2.

I D(G ) = 2(J − I −A(G )) +A(G ) = 2(J − I )−A(G ).I Since G is regular, J commutes with A(G ), and eigenvalue n

of J and k of G have a common eigenvector 1.

spec(D(G )) = {2n − 2− ρ} ∪{−2− λ : λ ∈ spec(A(G )) and λ 6= ρ}

= {2n − 2− k , (−2− θ)(mθ), (−2− τ)(mτ )}= {ρD, θ(mθ)D , τ

(mτ )D }.


Conference graphs are optimistic

A strongly regular graph is a conference graph if(n, k, λ, µ) = (n, n−12 ,

n−54 ,

n−14 ), or equivalently, if mθ = mτ

Theorem (Azarija; 2014)

A conference graph of order n is optimistic if and only if n ≥ 13.


Optimistic strongly regular graphs

spec(D(G )) = {2n − 2− k, (−2− θ)(mθ), (−2− τ)(mτ )}= {ρD, θ(mθ)D , τ

(mτ )D }.

Theorem (AABCDGHHKLT; 2016)

Let G be a strongly regular graph with parameters (n, k , λ, µ). Thegraph G is optimistic if and only if τD > 0 and mτ ≥ mθ. That is,G is optimistic if and only if

µ− 2kn − 1

≤ λ < −4 + µ+ k2

.


Complements of strongly regular graphs

If G is strongly regular, then G is strongly regular.


Let G be a strongly regular graph. Both G and G are optimistic ifand only if G is a conference graph and n ≥ 13.


Optimistic strongly regular graphs with λ = µ

G is optimistic if and only if τD > 0 and mτ ≥ mθ. That is, G isoptimistic if and only if

µ− 2kn − 1

≤ λ < −4 + µ+ k2

.

Corollary (AABCDGHHKLT; 2016)

A strongly regular graph with parameters (n, k, µ, µ) is optimistic ifand only if k > µ+ 4.


Other families of optimistic strongly regular graphs

A strongly regular graph with parameters (n, k, µ, µ) is optimistic ifand only if k > µ+ 4.

I The symplectic graph Sp(2m, q) is a strongly regular graphwith parameters

(n, k , λ, µ) =(q2m−1q−1 , q

2m−1, q2m−2(q − 1), q2m−2(q − 1))

I For m ≥ 2 and e = ±1, the graph O2m+1(3) on one type ofnonisotropic points is a strongly regular graph with parameters(

3m(e+3m)2 ,

3m−1(3m−e)2 ,

3m−1(3m−1−e)2 ,

3m−1(3m−1−e)2

).

Corollary (AABCDGHHKLT, 2016)

Sp(2m, q) is optimistic except for q = 2 and m = 2.O2m+1(3) is optimistic.


(B) Distance regular graphs with one positive distance eigenvalue

I G = (V ,E ) is distance regular if for u, v ∈ V withd(u, v) = k , the number of vertices w ∈ V such thatd(u,w) = i and d(v ,w) = j is independent of the choice of uand v .

I Distance regular graphs with exactly one positive distanceeigenvalue (counted according to multiplicity) are of particularinterest.

I [Koolen and Shpectorov, 1994] determined the distanceregular graphs with exactly one positive distance eigenvalue(counted according to multiplicity).

I Distance spectra of specific graphs can be determined bycomputation.

I Distance spectra of families are determined theoretically.

I This determination is now complete.


Distance regular graphs with one positive distanceeigenvalue: Specific graphs

KS # Graph G specD(G )

(II) Gosset graph {84, 0(48), (−12)(7)}(III) Schläfli graph {36, 0(20), (−6)(6)}(VI) Chang graphs (3) {42, 0(20), (−6)(7)}(IX) icosahedral graph

{18, 0(5), (−3 +

√5)(3) ,

(−3−√

5)(3)}

(XII) Petersen graph {15, 0(4), (−3)(5)}(XIII) dodecahedral graph

{50, 0(9), (−7 + 3

√5)(3),

(−2)(4), (−7− 3√

5)(3)}

.


Distance regular graphs with one positive distanceeigenvalue: Families of graphs

(I) Cocktail party graphs CP(m) = K2,2,...,2, strongly regular withparameters (2m, 2m − 2, 2m − 4, 2m − 2).specD(CP(m)) = {2m, 0(m−1),−2(m)}.

(X) Cycles: specD(Cn) =

I

{n2−14 , (−

14 sec

2(πjn ))(2), j = 1, . . . , p

}, for n = 2p + 1;

I

{n2

4 , 0(p−1), (− csc2(π(2j−1)n ))

(2), j = 1, . . . , b p2 c}

(∪ {−1}if p is odd), for n = 2p.

[Fowler, Caporossi, Hansen; 2001], [Atik, Panigrahi; 2015]



I Hamming graphs H(d , n) = Kn � · · · � Kn.I Hypercube Qd = H(d , 2).

I Halved cube 12Qd = Q2d−1.

(VII) Hamming graphs: specD(H(d , n)) ={dnd−1(n − 1), 0(nd−d(n−1)−1), (−nd−1)(d(n−1))

}.

[Indulal; 2009]

(IV) Halved cube: specD(12Qd) ={

d2d−3, 0(2d−1−(d+1)), (−2d−3)(d)

}.

[Atik, Panigrahi; 2016]



(VIII) Doob graph D(m, d):{3(2m + d)42m+d−1, 0(4

2m+d−6m−3d−1), (−42m+d−1)(6m+3d)}.

[AABCDGHHKLT; 2016]

(V) Johnson graphs

specD((J(n, r)) =

{s(n, r), 0((

nr)−n),

(− s(n,r)n−1

)(n−1)}where s(n, r) =

∑rj=0 j

(rj

)(n−rj

)[Atik, Panigrahi; 2015]

(XI) double odd graphs

specD(DO(r)) ={

(2r+1)(2r+1

r

), 0(2(

2r+1r )−2r−2),(

−2s(2r+1,r)r)(2r)

,−(2r+1)(2r+1

r

)+ 4s(2r+1, r)

}[AABCDGHHKLT; 2016]


(C) Distance determinants and inertias of generalizedbarbells and lollipops: Definitions

I For k ≥ 2, ` ≥ 0, a lollipop graph L(k, `) has a k-cliqueattached by an edge to one end of a path on ` vertices.

I For k ≥ 2, ` ≥ 0, a barbell graph B(k, `) has 2 k-cliques eachattached by an edge to one of the end vertices of a path on `vertices.

L(3, 2) B(4, 1)


Generalized barbells

I For k ,m ≥ 2, ` ≥ 0, a generalized barbell graph, B(k;m; `),has a k-clique attached by an edge to one end of a path on `vertices, and an m-clique attached by an edge to the otherend.

I L(k , `) = B(k ; 2; `− 2) for ` ≥ 2.I B(k , `) = B(k; k ; `) for k ≥ 2 and ` ≥ 0.

B(3; 2; 0) B(4; 4; 1)


Distance determinants and inertias of generalized barbells


detD(B(k ;m; `)) = (−1)k+m+`−12`(km(`+ 5)− 2(k + m)).


Inertia of D(B(k ;m; `)): (n+, n0, n−) = (1, 0, k + m + `− 1).


Distance determinants and inertias of generalized barbells:Examples

L(3, 2) = B(3, 2, 0) B(4, 1) = B(4, 4, 1)

I detD(B(3; 2; 0)) = 20.I specD(B(3; 2; 0)) ={−1,−4.26261,−1.17742,−0.568579, 7.00861}.

I detD(B(4; 4; 1)) = 160.I specD(B(4; 4; 1)) ={−11.2915,−2, (−1)(4),−0.708497,−0.539392, 18.5394

}


(D) Number of distinct distance eigenvalues:Few distinct distance eigenvalues and not distance regular

Question (Atik, Panigrahi; 2015; Problem 4.3)

Is there a graph G with less than diam(G ) + 1 distinct distanceeigenvalues and G is not distance regular?

Example (AABCDGHHKLT, 2016)

Q`d is not regular. diam(Q`d) = d + 1 but Q

`d has at most 5

distinct distance eigenvalues since its distance eigenvalues interlacethose of Qd , which has 3 distinct distance eigenvalues.


Few distinct distance eigenvalues and many degrees

If G is a transmission regular graph of order n withspecD(G ) = {ρ, θ2, . . . , θn}, then

specD(G �G ) = {2nρ} ∪ {(nθ2)(2), . . . , (nθn)(2)} ∪ {0((n−1)2)}.

Example (AABCDGHHKLT; 2016)

H

H2k

:= H2k−1�H2

k−1has k + 1 degrees and 5 distinct distance

eigenvalues.


Bound on number of distinct distance eigenvalues in a tree

I q(G ) is the number of distinct (adjacency) eigenvalues of G .

I qD(G ) is the number of distinct distance eigenvalues of G .

I It is well known that for a tree T ,

diam(T ) + 1 ≤ q(T ).

Conjecture (AABCDGHHKLT; 2016)

For a tree T ,diam(T ) + 1 ≤ qD(T ).


For a tree T , ⌈diam(T )

2

⌉≤ qD(T ).


[Merris, 1990] For T a tree, L(T ) is the line graph of T , A(L(T ))is the adjacency matrix of L(T ), and K := 2I +A(L(T )). Theeigenvalues of −2K−1 interlace the eigenvalues of D(T ).

Proof of⌈diam(T )

2

⌉≤ qD(T ).

I diam(L(T )) = diam(T )− 1.I A(L(T )) ≥ 0 has at least diam(L(T )) + 1 = diam(T ) distinct

eigenvalues.

I −2K−1 also has at least diam(T ) distinct eigenvalues.I The eigenvalues of −2K−1 interlace the eigenvalues of D.I Thus D has at least ddiam(T )2 e distinct eigenvalues.


Still many interesting open questions

Conjecture (Collins; 1989; attributed to Shor)

For every tree T of order at least 3, the location of the peak of theunimodal sequence of normalized distance coefficients

d0(T ), . . . , sn−2(T ) is between⌊n2

⌋and

⌈n − n√

5

⌉.

Question

What is the distance spectrum of the generalized barbellB(k ;m; `)?

Conjecture (AABCDGHHKLT; 2016)

For a tree T ,diam(T ) + 1 ≤ qD(T ).


Acknowledgements

The annual program Discrete Structures: Analysis andApplications at the Institute for Mathematics and its Applications(IMA) built connections that together with and Ben Braun’schapter in Recent Trends in Combinatorics were essential tosolving the Graham-Lovász Unimodality Conjecture.


Graduate Research Workshop in Combinatorics

This research took place at GRWC at Iowa State University,June 1-12, 2015. https://sites.google.com/site/rmgpgrwc/

Thanks to our sponsors who made this possible:I National Science Foundation DMS 1500662I ElsevierI Institute for Mathematics and Its ApplicationsI International Linear Algebra SocietyI Iowa State U, U Colorado-Denver, U Denver, U Nebraska, U

WyomingLeslie Hogben (Iowa State University and American Institute of Mathematics) 48 of 53

https://sites.google.com/site/rmgpgrwc/

References (Introduction)

R.L. Graham and H.O. Pollak. On the addressing problem forloop switching. Bell Syst. Tech. J. 50 (1971), 2495–2519.

M. Aouchiche and P. Hansen. Distance spectra of graphs: Asurvey. Linear Algebra Appl., 458 (2014), 301–386.

H. Hosoya, M. Murakami, M. Gotoh, Distance polynomial andcharacterization of a graph, Natur. Sci. Rep. OchanomizuUniv. 24 (1973) 27–34.

R.L. Graham, A.J. Hoffman, H. Hosoya, On the distancematrix of a directed graph, J. Graph Theory 1 (1977) 85–88.

M. Edelberg, M.R. Garey, R.L. Graham, On the distancematrix of a tree, Discrete Math. 14 (1976) 23–39.

R.L. Graham and L. Lovász. Distance matrix polynomials oftrees. Adv. Math. 29 (1978), 60–88.


References (Introduction continued)

R. Merris. The distance spectrum of a tree. J. Graph Theory14 (1990), 365–369.

G. Indulal. Distance spectrum of graphs compositions. ArsMathematica Contemporanea 2 (2009), 93–110.


References (Unimodality Conjecture)

G. Aalipour, A. Abiad, Z. Berikkyzy, L. Hogben, F.H.J.Kenter, J.C.-H. Lin, and M. Tait. Proof of a conjecture ofGraham and Lovász concerning unimodality of coefficients ofthe distance characteristic polynomial of a tree.arxiv:1507.02341[math.CO]

R.L. Graham and L. Lovász. Distance matrix polynomials oftrees. Adv. Math. 29 (1978), 60–88.

K.L. Collins. On a conjecture of Graham and Lovász aboutdistance matrices. Discrete Appl. Math. 25 (1989), 27–35.

B. Braun. Unimodality Problems in Ehrhart Theory. In RecentTrends in Combinatorics, IMA Volume, Springer, 2016.


arxiv:1507.02341 [math.CO]

References (On Distance Spectra)

G. Aalipour, A. Abiad, Z. Berikkyzy, J. Cummings, J. De Silva,W. Gao, K. Heysse, L. Hogben, F.H.J. Kenter, J.C.-H. Lin,and M. Tait. On the distance spectra of graphs. Linear AlgebraAppl. 497 (2016) 66–87.

J. Azarija. A short note on a short remark of Graham andLovász. Discrete Math. 315 (2014), 65–68.

J.H. Koolen and S.V. Shpectorov, Distance-regular Graphs theDistance Matrix of which has Only One Positive Eigenvalue.Europ. J. Combinatorics 15 (1994), 269–275.

P. W. Fowler, G. Caporossi, and P. Hansen. Distance matrices,Wiener indices, and related invariants of fullerenes. J. Phys.Chem. A, 105 (2001), 6232–6242.


References (On Distance Spectra continued)

F. Atik and P. Panigrahi. On the distance spectrum of distanceregular graphs, Linear Algebra Appl., 478 (2015), 256–273.

F. Atik and P. Panigrahi. Graphs with few distinct distanceeigenvalues irrespective of the diameters, Electronic J. ofLinear Algebra, 29 (2016), 194–205.


I. Introduction(A) Data communication problem: Loop switching(B) Distance matrices(C) Distance spectra results

II. Unimodality of distance characteristic polynomial coefficients for trees(A) Unimodality and Peak Conjectures of Graham and Lovász(B) Collins' counterexample to the Peak Conjecture(C) Proof of the Graham/Lovász Unimodality Conjecture(D) Collins/Shor Peak Conjecture(E) Graphs that are not trees

III. On distance spectra(A) Optimistic strongly regular graphs(B) Distance regular graphs having one positive distance eigenvalue(C) Distance determinants and inertias of generalized barbells and lollipops(D) Number of distinct distance eigenvalues

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Distance Spectra of Graphs - Iowa State University › lhogben › research › Hogben...Distance Spectra of Graphs Leslie Hogben Iowa State University and American Institute of Mathematics