2015/7/11, Chennai India, Quasi-spectral characterizations of graphs, (The Fourth India-Taiwan Conference on Discrete Mathematics, July 10-13, 2015)

Quasi-spectral characterizations of graphs

Chih-wen Weng

Department of Applied Mathematics National Chiao Tung University

July 10-13, 2015

Introduction

Theadjacency matrixA = (a_ij) of G is a binary square matrix of order n with rows and columns indexed by the vertex set VG of G such that for any i, j∈ VG, 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



 .

Thespectral radius ρ(G) of G is the largest eigenvalue of the adjacency matrix A of G.

Let G be a simple connected graph of n vertices and e edges with degree sequence d₁≥ d2 ≥ · · · ≥ dn. The relation between spectral radius ρ(G) and degree sequence d_i (or the number e) of G has been studied by many authors.

It is well-known that ρ(G)≤ d1 with equality if and only if G is regular [1988Minc, Chapter 2].

In 1985 [2002BH, Corollary 2.3], Brauldi and Hoffman showed that if e≤ k(k − 1)/2 then ρ(G) ≤ k − 1 with equality if and only if G ∼= Kn. In 1987 [1987S], Stanley showed that ρ(G)≤ ^−1+√₂^1+8e with equality if and only if G ∼= Kn.

In 1998 [1998H, Theorem 2], Yuan Hong showed that ρ(G)≤√

2e− n + 1 with equality if and only if G ∼= K1,n−1 or G ∼= Kn.

In 2001 [2001HSK, Theorem 2.3], Hong et al. showed that ρ(G)≤ ^dn−1+√

(dn+1)²+4(2e−ndⁿ)

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

In 2004 [2004SW, Theorem 2.2], Jinlong Shu and Yarong Wu showed that ρ(G)≤ ^dℓ−1+√

(dℓ+1)²+4(ℓ−1)(d1−dℓ)

2 for 1≤ ℓ ≤ n, with equality if and only if G is regular or there exists 2≤ t ≤ ℓ such that

d₁ = dt−1 = n− 1 and dt= dn.

The special case ℓ = 2 of [2004SW, Theorem 2.2] is reproved [2011D].

All the above results can be realized a special case of the following result.

Theorem A

.Theorem (Theorem A, [2013LW]) ..

...

For 1≤ ℓ ≤ n,

ρ(G)≤ ϕℓ := d_ℓ− 1 +√

(d_ℓ+ 1)²+ 4∑_ℓ−1

i=1(d_i− dℓ) 2

with equality if and only if there exists 1≤ t ≤ ℓ such that G = Kt−1+ H for some (d_n− t + 1)-regular graph H.

r r

r r

r

K⁻₅ = K₃+ N₂

r r

r r r

r r

S₇= K_1,6 = K₁+ N₆

r r

r r r

r r

W₇= K₁+ C₆

Idea of the proof of Theorem A

Minimize the maximum row-sum of













x₁ 0

. ..

x_ℓ−1 1

. ..

0 1













−1

A













x₁ 0

. ..

x_ℓ−1 1

. ..

0 1













for x₁, . . . , xℓ−1 ≥ 1, and apply Perron-Frobenius Theorem.

We now assume that G is bipartite.

Applying the idea in the proof of Theorem A, we have a similar theorem for bipartite graph.

Let G be a simple bipartite graph with bipartition orders p and q, and corresponding degree sequences d₁ ≥ d2 ≥ · · · ≥ dpand

d^′₁ ≥ d^′₂≥ · · · ≥ d^′q. For 1≤ s ≤ p and 1 ≤ t ≤ q, let

X_s,t = d_sd^′_t+

s−1

∑

i=1

(di− ds) +

t−1

∑

j=1

(d^′_j− d^′t),

Y_s,t =

s−1

∑

i=1

(di− ds)·

t−1

∑

j=1

(d^′_j− d^′t).

.Theorem (Theorem B, [2015LW]) ..

...

For 1≤ s ≤ p and 1 ≤ t ≤ q, and the notations Xs,t and Y_s,t in previous page, the spectral radius ρ(G) of a bipartite graph G satisfies

ρ(G)≤ ϕs,t:=

vu utX^s,t+

√X²_s,t− 4Ys,t

2 .

Furthermore, if G is connected then the above equality holds if and only if there exists nonnegative integers s^′ < s and t^′ < t, and a biregular graph H of bipartition orders p− s^′ and q− t^′ respectively such that G = K_s′,t^′ + H.

r r

r r r

K⁵_2,3 = ⁵K_2,3 = K_1,2+ N_1,1

r r

r r r r

5K_2,4= K_1,1+ N_1,3

Theorem B looks complex, but it is useful.

As shown in [2015LW], the following are also special cases of Theorem B:

ρ(G) ≤ √

e, [2008BFP]

ρ(G) ≤ ϕ1,1 =

√

d₁d^′₁, [2001BZ]

ρ(G) ≤ ϕ1,q =

√e− (q − d1)d^′_q,

ρ(G) ≤ ϕp,1 =

√

e− (p − d^′₁)dp, ρ(G) ≤ ϕp,q=

vu

ut2e − (pd^p+ qd^′_q− dpd^′_q) +

√

(pdp+ qd^′_q− dpd^′_q)²− 4dpd^′_q(pq− e)

2 .

Conjecture C

A. Bhattacharya, S. Friedland and U.N. Peled [2008BFP] gave the Conjecture C below.

.Conjecture (Conjecture C) ..

...

LetK(p, q, e) denote the family of e-edge subgraphs of the complete bipartite graph K_p,qwith bipartition orders p and q, and 1 < e < pq be integers. An extremal graph that solves

max

G∈K(p,q,e)ρ(G)

is obtained from a complete bipartite graph by adding one vertex and a corresponding number of edges.

Extremal graphs in Conjecture C

For e≥ pq − max(p, q) (resp. e ≥ pq − min(p, q)), let ^eK_p,q (resp. K^e_p,q) denote the graph which is obtained from K_p,q by deleting pq− e edges incident on a common vertex in the partite set of order no larger than (resp. no less than) that of the other partite set. Then the extremal graph in Conjecture B is either ^eK_s,t or K^e_s,t.

r r

r r r

K⁵_2,3 = ⁵K_2,3

r r

r r r r

5K_2,4

ρ(K⁵_2,3)≥ ρ(⁵K_2,4) or ρ(⁵K_2,4)≥ ρ(K⁵2,3)?

Conjecture D

In 2010 [2010FKSW], Yi-Fan Chen, Hung-Lin Fu, In-Jae Kim, Eryn Stehr and Brendon Watts determined ρ(K^e_p,q) and gave an affirmative answer to Conjecture C when e = pq− 2.

Furthermore, they refined Conjecture C for the case when the number of edges is at least pq− min(p, q) + 1 to the following conjecture.

.Conjecture (Conjecture D) ..

...

Suppose 0 < pq− e < min(p, q). Then for G ∈ K(p, q, e), ρ(G)≤ ρ(K^ep,q).

Conjecture D is affirmative by an application of Theorem B [2015LW].

Average 2-degree sequence

Theaverage 2-degreeof the vertex v∈ VG is m(v) := 1

d(v)

∑

uv∈EG

d(u),

where d(u) is the degree of u∈ VG.

Let m₁ ≥ m2 ≥ · · · ≥ mn denote the sequence of average 2-degrees of G.

Theorem E

With similar idea of Theorrm A, we showed the following theorem.

.Theorem (Theorem E, [2014HW]) ..

...

For any b≥ max {d(i)/d(j) | ij ∈ EG} and 1 ≤ ℓ ≤ n,

ρ(G)≤ m_ℓ− b +√

(mℓ+ b)²+ 4b∑_l−1

i=1(mi− mℓ)

2 ,

with equality holds iff G is pseudo regular (i.e. m₁ = mn).

r r r r r r r r r r r r r r

r r r r r r r

r

Figure: A pseudo regular tree.

Let D(G)=diag(d(1), d(2),· · · , d(n)) be the diagonal matrix, where d(i) is degree of vertex i. Then the matrix

L(G) = D(G)− A(G) is called the Laplacian matrixof G.

.Example ..

...

d d d

1 2 3

Then

L =



 1 −1 0

−1 2 −1

0 −1 1



 .

We call the eigenvalues of L(G) the Laplacian eigenvaluesof G. It is well known that L(G) is symmetric, positive semidefinite, and every row sum being zero, so we denote the Laplacian eigenvalues of G as in

nonincreasing order as

ℓ₁(G)≥ ℓ2(G)≥ · · · ≥ ℓn(G) = 0.

TheLaplacian spread of G is defined as

SL(G) := ℓ1(G)− ℓn−1(G).

G is calledstrongly regularwith parameters (n, k, λ, µ) if G is a k-regular graph with order n and λ (resp. µ) common neighbors of any pair of two adjacent (resp. nonadjacent) vertices.

t t

t t

t

t t

t t

t

Figure: The Petersen graph is strongly regular with parameters (10, 3, 0, 1).

Let G be a simple connected graph with vertex set VG ={1, 2, · · · , n} and edge set EG. Define

λ_min(G) := min

ij∈EG|N(i) ∩ N(j)|;

µmin(G) := min

ij̸∈EG|N(i) ∩ N(j)|, where N(i) is the neighbor of vertex i∈ V.

Let δ, ∆ be minimum degree and maximum degree of G respectively.

.Theorem (Theorem F1, [LWpre]) ..

... ℓ1(G)≤ 2∆− λmin+ µmin+√

(2∆− λmin+ µmin)²− 4nµmin

2 .

..

6 .

5 .

4

.

3

.

2

. 1

. 8

. 7

.Theorem (Theorem F2, [LWpre]) ..

...

ℓn−1(G)

≥ 2δ− λmin+ µmin−√

(2δ− λmin+ µmin)²− 4nµmin− 4δ²+ 4∆²

2 .

..

6 .

5 .

4

.

3

.

2

. 1

. 8

. 7

.Theorem (Theorem F3, [LWpre]) ..

...

SL(G) ≤ ∆ − δ + 1 2

[√(2∆− λmin+ µmin)²− 4nµmin

+√

(2δ− λmin+ µmin)²− 4nµmin− 4δ²+ 4∆² ]

.

..

2 .

1 .

2t .

2t− 1 .

...

. 8

.

7

. ⁶

.

5 .

4 .

3 .

2t + 1

The matrix

Q(G) = D(G)− A(G)

is called the signless Laplacian matrixof G. Let q(G) denote the largest eigenvalue of Q(G).

.Example ..

...

d d d

1 2 3

Then

Q =



 1 1 0 1 2 1 0 1 1



 .

Let G be a simple connected graph with vertex set VG ={1, 2, · · · , n} and edge set EG. Define

λmax(G) := max

ij∈EG|N(i) ∩ N(j)|;

µmax(G) := max

ij̸∈EG|N(i) ∩ N(j)|.

.Theorem (Theorem G, [FRpre]) ..

...

q(G)≤ ∆ − µmax

4 +

√(

∆−µmax

4 )₂

+ ∆(1 + λmax) + (n− 1)µmax− ∆², with equality if and only if G is strongly regular graph with parameters (n, ∆, λmax, µmax).

Idea of the proof of Theorem G (Theorems F123 are similar)

Let X be the Perron eigenvector of Q(G). Using the properties that X^TQ(G^c)X ≤ 2n − 2 − q(G),

X^TQ(G^c)X = ∑

i<j,ij̸∈E

(xi+ xj)²,

inequalities and some combinatorial arguments to evaluate the term

∥A(G)X∥² = X^⊤A(G)²X =∑

i∈V

d_ix²_i + 2∑

j<k

(A(G)²)jkx_jx_k. (1)

in

∥(q(G)I − D(G))X∥² =∥(Q(G) − D(G))X∥² =∥A(G)X∥², and then obtained an equality with X involved. Try to find another inequality without X.

Future Project

Some graphs are characterized in terms of one of their eigenvalues and some combinatorial parameters.

(Quasi-spectral characterization)

If all eigenvalues of such a graph are provided, it might be possible to characterize the graph only by its eigenvalues.

(Spectral characterization)

References

[2001BZ] A. Berman and X.-D. Zhang, On the spectral radius of graphs with cut vertices, J. Combin. Theory Ser. B 83 (2001) 233-240.

[2008BFP] A. Bhattacharya, S. Friedland and U.N. Peled, On the first eigenvalue of bipartite graphs, Electron J. Combinatorics 15 (2008), ♯R144.

[2002BH] R.A. Brualdi and A.J. Hoffman, On the spectral radius of (0, 1)-matrices, Appl. Math. J. Ser. B 3 (2002) 371-376.

[2010FKSW] Y.-F. Chen, H.-L. Fu, I.-J. Kim, E. Stehr and B. Watts, On the largest eigenvalues of bipartite graphs which are nearly complete, Linear Algebra Appl. 432 (2010) 606-614.

[FRpre] Feng-lei Fan and Chih-wen Weng, A characterization of strongly regular graph in terms of the largest signless Laplacian eigenvalue, preprint.

[2011D] K.Ch. Das, Proof of conjecture involving the second largest signless Laplacian eigenvalue and the index of graphs, Linear Algebra Appl. 435 (2011) 2420-2424.

[1998H] Y. Hong, Upper bounds of the spectral radius of graphs in terms of genus, J.

Combin. Theory Ser. B 74 (1998) 153-159.

[2001HSK] Y. Hong, J. Shu and Kunfu Fang, A sharp upper bound of the spectral radius of graphs, J. Combin. Theory Ser. B 81 (2001) 177-183.

[2014HW] Y.-P. Huang and C.-W. Weng, Spectral radius and average 2-degree sequence of a graph, Discrete Math. Algorithm. Appl. Vol. 6 No. 2 (2014) 1450029.

[LWpre] Fan-Hsuan Lin and Chih-wen Weng, A bound on the Laplacian spread which is tight for strongly regular graphs, preprint

[2013LW] C.-A. Liu and C.-W. Weng, Spectral radius and degree sequence of a graph, Linear Algebra Appl. 438 (2013) 3511-3515.

[2015LW] C.-A. Liu and C.-W. Weng, Spectral radius of bipartite graphs, Linear Algebra and its Applications, 474(2015), 30–43

[1988Minc] Henryk Minc, Nonnegative Matrices, John Wiley and Sons Inc., New York, 1988.

[2004SW] J. Shu and Y. Wu, Sharp upper bounds on the spectral radius of graphs, Linear Algebra Appl. 377 (2004) 241-248.

[1987S] R.P. Stanley, A bound on the spectral radius of graphs with e edges, Linear Algebra Appl. 87 (1987) 267-269.