2013/7/12, 交大, Spectral characterization of graphs, (2013 International Conference on Combinatorics, 7/11-12, 2013)

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2013 International Conference on Combinatorics

Spectral characterization of graphs

Chih-wen Weng

Joint work with Yu-pei Huang, Guang-Siang Lee and Chia-an Liu

Department of Applied Mathematics National Chiao Tung University

July 12, 2013

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Notations

Let G be a simple connected graph of order n.

Theadjacency matrixA = (a_ij) of G is a binary square matrix of order n with rows and columns indexed by the vertex set V G of G such that for any i, j∈ V G, aij = 1 if i, j are adjacent in G.

d d d

1 2 3

A =



 0 1 0 1 0 1 0 1 0



 .

Let λ₁(A)≥ λ2(A)≥ · · · ≥ λn(A) denote the eigenvalues of A, and λ_i(G) := λ_i(A).

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Eigenvalues help us to realize the structure of a graph

.Theorem ..

...

For a graph G of order n, G is bipartite if and only if λ₁(G) =−λn(G).

Dongbo Bu, et al., Topological structure analysis of the protein-protein interaction network in budding yeast, Nucleic Acids Research, 2003, Vol.

31, No. 9, 2443-2450.

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Eigenvalues help us to solve problems in Combinatorics

Let χ(G) denote the chromatic number of G.

.Theorem (Wilf Theorem(1967) and Hoﬀman(1970)) ..

...

For a graph G,

(λ_n(G)− λ1(G))/λ_n(G)≤ χ(G) ≤ λ1(G) + 1.

To estimate the integer value χ(G), only approximations of λ₁(G) and λn(G) are necessary.

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Estimate the eigenvalues of a matrix by matrices of smaller sizes

It is well-known that λ1= max

x∈Rn x⊤x=1

x^⊤Ax, λn= min

x∈Rn x⊤x=1

x^⊤Ax.

The following theorem generalizes this property.

.Theorem (Cauchy interlacing theorem) ..

...

For m < n, and an m× n matrix S with SS^⊤= I, λi(A)≥ λi(SAS^⊤), λn+1−i(A)≤ λm+1−i(SAS^⊤) for 1≤ i ≤ m.

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Example

Choose S = [I 0] in block form and then SAS^⊤ becomes the adjacency matrix of an induced subgraph of G.

List the eigenvalues of paths P_n and P_n₋₁ of orders n and n− 1 respectively:

2 cos π

n + 1> 2 cos 2π

n + 1> 2 cos 3π

n + 1>· · · > 2 cos(n− 1)π

n + 1 > 2 cos nπ n + 1

↘ 2 cosπ

n> 2 cos2π

n > · · · > 2 cos(n− 1)π

n ↗

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The above method does not give us an upper bound of λ₁(A).

Can we ﬁnd a matrix M whose largest eigenvalue λ₁(M ) gives an upper bound of λ₁(G)?

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Perron-Frobenius Theorem

Let d₁ ≥ d2 ≥ · · · ≥ dn denote the degree sequence of G.

.Theorem ..

...

λ₁(G)≤ d1

with equality iﬀ G is regular.

Let [d₁] be a 1× 1 matrix. The above theorem says λ1(G)≤ λ1([d1]).

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Another upper bound of λ₁(G) is .Theorem (Stanley, 1987) ..

...

λ₁(G)≤ −1 +√

1 + 8|EG|

2

with equality if and only if G is the complete graph K_n.

Equivalently, λ₁(G) is bounded above by

λ1













0 1 · · · 1 d1− (n − 1) 1 0 1 · · · 1 d2− (n − 1)

... . .. ... ... ...

1 0 d_n− (n − 1)

1 · · · 1 dn+1− n







(n+1)×(n+1)





 ,

where d_n+1:= 0, thinking of an isolated vertex being added.

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An improvement of Stanley Theorem is

.Theorem (Yuan Hong, Jin-Long Shu and Kunfu Fang, 2001) ..

...

λ1(G)≤ dn− 1 +√

(dn+ 1)²+ 4(2|EG| − ndn)

2 ,

with equality if and only if G is regular or there exists 2≤ t ≤ n such that d₁ = d_t₋₁ = n− 1 and dt= d_n.

Equivalently, λ₁(G) is bounded above by

λ1













0 1 · · · 1 d1− n + 2 1 0 1 · · · 1 d2− n + 2

... . .. ... ... ...

1 0 d_n₋₁− n + 2 1 · · · 1 dn− n + 1







×n





 .

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Another version is

.Theorem (Kinkar Ch. Das, 2011) ..

...

λ₁(G)≤ d2− 1 +√

(d2+ 1)²+ 4(d1− d2)

2 ,

with equality if and only if either G is regular, or d₁ = n− 1 and d₂ = d_n.

Equivalently,

λ1(G)≤ λ1

([ 0 d₁ 1 d2− 1

]

2×2

) .

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The parameter ϕ

ℓ For 1≤ ℓ ≤ n, let

ϕ_ℓ(G) :=λ1













0 1 · · · 1 d₁− ℓ + 2 1 0 1 · · · 1 d2− ℓ + 2

... . .. ... ... ...

1 0 d_ℓ₋₁− ℓ + 2 1 · · · 1 dℓ− ℓ + 1







ℓ×ℓ







=d_ℓ− 1 +√

(d_ℓ+ 1)²+ 4∑_ℓ₋₁

i=1(d_i− dℓ)

2 .

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.Theorem (Chia-an Liu, —, 2013) ..

...

For each 1≤ ℓ ≤ n,

λ1(G)≤ ϕℓ(G),

with equality iﬀ G is regular or there exists 2≤ t ≤ ℓ such that d1 = dt−1 = n− 1 and dt= dn.

Moreover, we show that the function ϕ_ℓ(G) in variable ℓ is convex.

ϕ₁ _r ϕ2 r

r r r r

r ϕ_n r ϕn−1

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Small technical diﬃculty in the proof

The matrix







0 1 · · · 1 d₁− ℓ + 2 1 0 1 · · · 1 d2− ℓ + 2 ... . .. ... ... ...

1 0 d_ℓ−1− ℓ + 2 1 · · · 1 d_ℓ− ℓ + 1







ℓ×ℓ

needs not to be nonnegative.

Our formal proof follows the idea of Jinlong Shu and Yarong Wu 2004, which applies Perron-Frobenius Theorem to U⁻¹AU with some carefully selected diagonal matrix U .

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The number m_i:= _d¹

i

∑

j∼id_j is called the average 2-degreeof i. List m_i in the decreasing ordering as

M1 ≥ M2 ≥ · · · ≥ Mn.

t t

t

t t

A non-regular graph with M₁ = M2 =· · · = M9= 3

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Applying Perron-Frobenius Theorem to







d₁ 0

d2

. ..

0 d_n







−1

A







d₁ 0

d2

. ..

0 d_n





,

we have .Theorem ..

...

λ₁(G)≤ M1

with equality iﬀ M₁ = M_n.

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An improvement of the upper bound M₁,

.Theorem (Ya-hong Chen and Rong-yin Pan and Xiao-dong Zhang, 2011)

..

...

λ1(G)≤ M₂− a +√

(M₂+ a)²+ 4a(M₁− M2)

2 ,

with equality iﬀ M₁ = M_n, where a = max{di/d_j | 1 ≤ i, j ≤ n}.

Equivalently,

λ₁(G)≤ λ1

([0 M1

a M₂− a ])

.

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Let b≥ max{di/d_j | 1 ≤ i, j ≤ n, i ∼ j}, and for 1 ≤ ℓ ≤ n, let

ψ_ℓ(G) :=λ₁













0 b · · · b M₁− (ℓ − 2)b b 0 b · · · b M2− (ℓ − 2)b ... . .. ... ... ...

b 0 Mℓ−1− (ℓ − 2)b b · · · b M_ℓ− (ℓ − 1)b







ℓ×ℓ







=Mℓ− b +√

(Mℓ+ b)²+ 4b∑_ℓ₋₁

i=1(Mi− Mℓ)

2 .

.Theorem (Yu-pei Huang, —, 2013) ..

...

For each 1≤ ℓ ≤ n,

λ1(G)≤ ψℓ(G), with equality iﬀ M₁ = M_n.

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Problem: In the spirit of Cauchy interlacing theorem, give a uniform way to ﬁnd a matrix M with λ₁(A)≤ λ1(M ) that generalizes the above matrices.

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Sometimes, the eigenvector α > 0 (Perron vector) of A corresponding to λ1(A) also involves in the study.

For instance the Perron vector of the web graph plays a key role in ranking the web pages by Google.

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Our second spectral characterization of graphs is related to distance-regular graphs.

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Distance-regular graphs

We recall deﬁnition of DRGs and their basic properties.

A graph G with diameter D isdistance-regular if and only if for i≤ D, ci := |G1(x)∩ Gi−1(y)|,

a_i := |G1(x)∩ Gi(y)|, b_i := |G1(x)∩ Gi+1(y)| are constantssubject to all vertices x, y with ∂(x, y) = i.

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∂(x, y) = i

d y

d x

ci

ai

bi

Note that a_i+ bi+ ci= b0 and k := b₀ is the valency of G.

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Distance-Regular graphs, also called P -polynomial schemes, form an important subclass of association schemes.

”Association schemes are the frameworks on which coding theory, design theory and other theories developed in a uniﬁed and satisfactory way. ...

There are many mathematical objects whose essence is that of association schemes and many diﬀerent names are given to the essentially the same mathematical concept: Adjacency algebra, Bose-Mesner algebra,

centralizer ring, Hecke ring, Schur ring, character algebra, hypergroup, probabilistic group, etc” ——–Eiichi Bannai and Tatsuro Ito

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Distance matrices

The matrices that we are concerned are square matrices with rows and columns indexed by the vertex set V G. Let α be an eigenvector of A corresponding to λ₁(G) normalized to α^⊤α = n. For each i let Ai be the matrix with entries

(Ai)xy =

{ α_xα_y, if ∂(x, y) = i;

0, else.

A_i is called i-th distance matrixof Γ. Note A₀ = I and A₋₁= A_D+1= 0.

If G is regular then α = (1, 1, . . . , 1)^⊤, so A_i is binary and A₁= A.

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t t t

t

t t t

t G

A0 = I,

A1 =







0 0 0 0 1 1 1 0 0 0 0 0 1 1 0 1 0 0 0 0 1 0 1 1 0 0 0 0 0 1 1 1 1 1 1 0 0 0 0 0 1 1 0 1 0 0 0 0 1 0 1 1 0 0 0 0





 ,

A₂ =







0 1 1 1 0 0 0 0 1 0 1 1 0 0 0 0 1 1 0 1 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 1 0 1 1 0 0 0 0 1 1 0 1 0 0 0 0 1 1 1 0





 ,

A3 =







0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0





 .

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Three-term recurrence relation of DRGs

.Theorem ..

...

Let G be a regular graph. Then the following are equivalent.

.

1.. G is distance-regular;

.

2.. AAi = bi−1Ai−1+ aiAi+ ci+1Ai+1 0≤ i ≤ D;

.

3.. there exist a unique sequence of polynomials p₀(x) = 1, p1(x) = x, . . ., pD(x) such that deg(pi) = i and Ai= pi(A).

The polynomials p₀(x) = 1, p₁(x) = x, . . ., p_D(x) are calleddistance polynomials of a DRG, but they can be reconstructed in a general graph.

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Let G a general graph G with adjacency matrix A and minimal polynomial of degree d + 1. Since A is symmetric, A has d + 1 distinct eigenvalues.

The number d is called thespectral diameterof G. It is well-known that d≥ D.

Deﬁne an inner product on the space of real polynomials of degrees at most d by

⟨f(λ), g(λ)⟩ = 1

ntrace(

f (A)g(A)^⊤ )

.

Then there exists a unique sequence of orthogonal polynomials p₀(x) = 1, p1(x), . . ., pd(x) such that

deg(p_i) = i, and ⟨pi(x), pi(x)⟩ = pi(λ1).

G is t-partially distance-regular if A = p (A) for 0≤ i ≤ t.

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The number

pd(λ1)

is called the spectral excess ofG; while the number δ_D := 1

ntrace(A_DA^⊤_D) is called the excessof G.

When G is regular

δD = 1 n

∑

x∈V (G)

|GD(x)|

is the average number of vertices which have distance the diameter to a vertex.

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Spectral Excess Theorem

.Theorem (M.A. Fiol, E. Garriga and J.L.A. Yebra, 1996) ..

...

If G is regular then

δ_D ≤ pd(λ₁), with equality iﬀ G is distance-regular.

Short proofs are given by [E.R. van Dam, 2008] and [M.A. Fiol, S. Gago and E. Garriga, 2010].

Base on the short proofs, the regularity assumption of G is dropped in the Spectral Excess Theorem by [Guang-Siang Lee, , 2012].

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Application

Theodd girth of a graph is the smallest length of an odd cycle in the the graph.

.Corollary (E.R. van Dam and W.H. Haemers, 2011) ..

...A regular graph with odd girth 2d + 1 is a generalized odd graph.

The above corollary generalizes the spectral characterization of generalized odd graphs [Tayuan Huang, 1994], [Tayuan Huang and Chao Rong Liu, 1999]. Tayuan Huang is an Emeritus of NCTU.

The regularity assumption is dropped in the above corollary by [Guang-Siang Lee, , 2012].

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Applying Spectral Excess Theorem to bipartite graphs, we have .Theorem (Guang-Siang Lee, , 2013)

..

...

Assume G is bipartite with bipartition X∪ Y andevenspectral diameter d.

Then the following are equivalent.

(i) δ_D = p_d(λ₁);

(ii) G is distance-regular;

(iii) G is 2-partially distance-regular and both of the halved graphs G^X and G^Y are distance-regular.

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The assumption 2-partially distance-regular is necessary

The following example gives a regular bipartite graph G with G^X = G^Y being a clique and even spectral diameter, but G is not 2-partially distance-regular.

.Example ..

...

Let G = K_5,5− C4− C6 be a regular graph obtained by deleting a C₄ and a C₆ from K_5,5. We have sp G ={3¹, 2¹, 1², 0², (−1)², (−2)¹, (−3)¹}, D = 3 < 6 = d and G²= 2K₅.

t t t t t

C₄+ C₆

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.Example ..

...

Let G be the Hoﬀman graph, which is a cospectral graph of 4-cube obtained from 4-cune by applying GM-swithching of edges. Then sp G ={4¹, 2⁴, 0⁶, (−2)⁴, (−4)¹}, D = d = 4, and

A_i = p_i(A) iﬀ i∈ {0, 1, 3}.

Note that G² is the disjoint union of K₈ and K_2,2,2,2(= K8− 4K2), which are both distance-regular (sp K_2,2,2,2={6¹, 0⁴, (−2)³}).

The 4-cube. The Hoﬀman graph.

Copy from http://en.wikipedia.org/wiki/Hoﬀman_graph

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Another drawing of 4-cube and Hoﬀman graph

The 4-cube q

q

q q

q q q q

The Hoﬀman graph

x D

q

q q

q q q q

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The assumption even spectral diameter is necessary

The following example gives a bipartite 2-partially distance-regular graph G with D = d = 5 such that G^X, G^Y are distance-regular graphs with spectrum{6¹, 1⁴, (−2)⁵} (the complement of petersen graph), but G is not distance-regular.

.Example ..

...

Consider the regular bipartite graphs G on 20 vertices obtained from the Desargues graph (the bipartite double of the Petersen graph) by the GM-switching. One can check (by Maple) that D = d = 5,

sp G ={3¹, 2⁴, 1⁵, (−1)⁵, (−2)⁴, (−3)¹}, and

Ai= pi(A) iﬀ i∈ {0, 1, 2, 4}.

Then G is not distance-regular.

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Desargues graph and its cospectral mate

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Near DRGs

Similar to the deﬁnition of excess, one can deﬁne δ_i:= 1

ntrace(A_iA^⊤_i ),

and want to characterize the graphs satisfying δ_i = pi(λ1) for some i.

Note that

A_i = p_i(A) ⇒ δi= p_i(λ₁).

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A bipartite graph with bipartitin V (G) = X∪ Y is biregularif there exist distinct integers k̸= k^′ such that every x∈ X has degree k, and every y∈ Y has degree k^′.

.Proposition ..

...

Let G be a connected graph. Then δ₁ ≥ p1(λ₁), and the following statements are equivalent.

(i) δ₁ = p1(λ1), (ii) A₁ = p₁(A),

(iii) G is regular or G is bipartite biregular.

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.Theorem (Guang-Siang Lee, , 2013) ..

...

Let G be a connected bipartite graph with bipartition X∪ Y and assume that the spectral diameter d is odd. Then the following are equivalent.

(i) δi = pi(λ1) for even i;

(ii) δd−1 = pd−1(λ1);

(iii) G is 2-partially distance-regular and both of the halved graphs G^X and G^Y are distance-regular ⌊d/2⌋.

We shall provide two graphs that satisfy the above equivalent conditions, but are not distance-regular graphs.

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We saw the ﬁrst one before.

.Example (W.H. Haemers and E. Spence, 1995) ..

...

Consider the regular bipartite graphs G on 20 vertices obtained from the Desargues graph (the bipartite double of the Petersen graph) by the GM-switching. One can check (by Maple) that D = d = 5,

sp G ={3¹, 2⁴, 1⁵, (−1)⁵, (−2)⁴, (−3)¹}, and

Ai= pi(A) iﬀ i∈ {0, 1, 2, 4}.

Then G is not distance-regular.

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Desargues graph and its cospectral mate

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.Example (D. Marušič and T. Pisanski, 2000) ..

...

Consider the Möbius-Kantor graph G. One can check (by Maple) that D = 4 < 5 = d, and

A_i = p_i(A) iﬀ i∈ {0, 1, 2, 4}.

Möbius-Kantor graph Copy from https://en.wikipedia.org/wiki

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References I

.

1.. Ya-hong Chen and Rong-yin Pan and Xiao-dong Zhang, Two sharp upper bounds for the signless Laplacian spectral radius of graphs, Discrete Mathematics, Algorithms and Applications, Vol. 3, No. 2 (2011), 185-191.

.

2.. Kinkar Ch. Das, Proof of conjecture involving the second largest signless Laplacian eigenvalue and the index of graphs, Linear Algebra and its Applications, 435 (2011), 2420-2424.

.

3.. Yuan Hong, Jin-Long Shu and Kunfu Fang, A sharp upper bound of the spectral radius of graphs, Journal of Combinatorial Theory, Series B 81 (2001), 177-183.

.

4.. Jinlong Shu and Yarong Wu, Sharp upper bounds on the spectral radius of graphs, Linear Algebra and its Applications, 377 (2004), 241-248.

.

5.. Richard. P. Stanley, A bound on the spectral radius of graphs with e edges, Linear Algebra and its Applications, 87 (1987), 267-269.

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References II

.

1.. E.R. van Dam, The spectral excess theorem for distance-regular graphs: a global (over)view, Electron. J. Combin. 15 (1) (2008),

#R129.

.

2.. E.R. van Dam and W.H. Haemers, An odd characterization of the generalized odd graphs, J. Combin. Theory Ser. B 101 (2011), 486-489.

.

3.. M.A. Fiol, E. Garriga and J.L.A. Yebra, On a class of polynomials and its relation with the spectra and diameters of graphs, J. Combin.

Theory Ser. B 67 (1996), 48-61.

.

4.. M.A. Fiol, S. Gago and E. Garriga, A simple proof of the spectral excess theorem for distance-regular graphs, Linear Algebra and its Applications, 432(2010), 2418-2422.

.

5.. W.H. Haemers and E. Spence, Graphs cospectral with distance-regular graphs, Linear Multili. Alg. 39 (1995), 91-107.

.

6.. D. Marušič and T. Pisanski, The remarkable generalized Petersen graph G(8, 3), Math. Slovaca 50 (2000), 117-121.

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References III

.

1.. Yu-pei Huang and Chih-wen Weng, Spectral Radius and Average 2-Degree Sequence of a Graph, preprint.

.

2.. Guang-Siang Lee and Chih-wen Weng, A spectral excess theorem for nonregular graphs, Journal of Combinatorial Theory, Series A, 119(2012), 1427-1431.

.

3.. Guang-Siang Lee and Chih-wen Weng, A characterization of bipartite distance-regular graphs, preprint.

.

4.. Chia-an Liu and Chih-wen Weng, Spectral radius and degree sequence of a graph, Linear Algebra and its Applications, (2013), http://

dx.doi.org/10.1016/j.laa.2012.12.016

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2013/7/12, 交大, Spectral characterization of graphs, (2013 International Conference on Combinatorics, 7/11-12, 2013)

Spectral characterization of graphs

Notations

Eigenvalues help us to realize the structure of a graph

Eigenvalues help us to solve problems in Combinatorics

Estimate the eigenvalues of a matrix by matrices of smaller sizes

Example

Perron-Frobenius Theorem

The parameter ϕ

Small technical diﬃculty in the proof

Distance-regular graphs

∂(x, y) = i

Distance matrices

Three-term recurrence relation of DRGs

Spectral Excess Theorem

Application

The assumption 2-partially distance-regular is necessary

Another drawing of 4-cube and Hoﬀman graph

The assumption even spectral diameter is necessary

Desargues graph and its cospectral mate

Near DRGs

Desargues graph and its cospectral mate

References I

References II

References III

Thanks for your attention.