A characterization of strongly regular graphs in terms of the largest signless Laplacian eigenvalues

Linear Algebra and its Applications 506 (2016) 1–5

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Linear Algebra and its Applications www.elsevier.com/locate/laa

A characterization of strongly regular graphs in terms of the largest signless Laplacian eigenvalues Feng-lei Fan a,∗ , Chih-wen Weng b a

Department of Automatic Test and Control, Harbin Institute of Technology, 92 Xidazhi Street, Harbin, China b Department of Applied Mathematics, National Chiao Tung University, 1001 Ta Hsueh Road, Hsinchu, Taiwan, ROC

a r t i c l e

i n f o

Article history: Received 28 May 2015 Accepted 9 May 2016 Available online xxxx Submitted by R. Brualdi MSC: 05C50 05E30 15A42 Keywords: Signless Laplacian matrix Strongly regular graphs

a b s t r a c t Let G be a simple graph of order n with maximum degree Δ. Let λ (resp. μ) denote the maximum number of common neighbors of a pair of adjacent vertices (resp. nonadjacent distinct vertices) of G. Let q(G) denote the largest eigenvalue of the signless Laplacian matrix of G. We show that μ q(G) ≤ Δ − + 4



Δ−

μ 2 + (1 + λ)Δ + μ(n − 1) − Δ2 , 4

with equality if and only if G is a strongly regular graph with parameters (n, Δ, λ, μ). © 2016 Elsevier Inc. All rights reserved.

1. Introduction Let G be a simple graph of order n with maximum degree Δ. Let λ (resp. μ) denote the maximum number of common neighbors of a pair of adjacent vertices (resp. nonadjacent distinct vertices) of G. Then G is called strongly regular with parameters (n, Δ, λ, μ) * Corresponding author. E-mail addresses: [email protected] (F.-l. Fan), [email protected] (C.-w. Weng). http://dx.doi.org/10.1016/j.laa.2016.05.009 0024-3795/© 2016 Elsevier Inc. All rights reserved.

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F.-l. Fan, C.-w. Weng / Linear Algebra and its Applications 506 (2016) 1–5

if G has order n, every vertex of G has the same degree Δ, and every chosen pair of adjacent vertices (resp. nonadjacent distinct vertices) has the same number λ (resp. μ) of common neighbors. Strongly regular graphs receive attentions from many different areas of researchers. For example, strongly regular graphs form the simplest nontrivial class of distance-regular graphs [2], or more generally of association schemes [1]. Any regular connected graph with three distinct eigenvalues must be strongly regular graphs [3, Theorem 9.1.2]. See [12, Chapter 21] for many beautiful properties of strongly regular graphs. This note aims to characterize strongly regular graphs in terms of the graph parameters n, Δ, λ, μ and an additional parameter q(G), the largest eigenvalue of the signless Laplacian matrix associated with G. Theorem 3.1 is our main result. 2. Preliminaries In this section we shall introduce notations and basic properties that are needed in the statement and proof of our main result. For more reading, the reader may refer to [3]. Throughout this note, let G = (V, E) be a simple graph of order n with vertex set V = {1, 2, . . . , n} and edge set E. The adjacency matrix A(G) of G is an n-by-n 01-matrix whose ij-entry is 1 or 0 according to whether i and j are adjacent or not adjacent respectively. Then the parameters λ and μ defined in the previous section can be restated as λ=

max (A(G)2 )jk ,

j=k,jk∈E

μ=

max (A(G)2 )jk .

j=k,jk∈E /

The degree di of a vertex i ∈ V is the number |N (i)|, where N (i) is the set of neighbors of i. The number Δ = max1≤i≤n di is called the maximum degree of G. Note that μ = 0 implies that Δ = λ + 1, and G is a union of cliques, among them the maximal one has order Δ + 1. Let ρ(A(G)) denote the largest eigenvalue of A(G). Then the Rayleigh principle implies that xT A(G)x ≤ ρ(A(G))xT x, where x is a column vector of length n. By the Perron–Frobenius Theorem, ρ(A(G)) ≤ Δ [13, Chapter 2]. Let D(G) = diag(d1 , d2 , . . . , dn ) be the diagonal matrix with entries d1 , d2 , . . . , dn in the diagonal. Then the matrices Q(G) = D(G) +A(G) and L(G) = D(G) −A(G) are called the signless Laplacian matrix and Laplacian matrix of G respectively. It was checked in small graphs by D. Cvetković and S.K. Simić, and they found that less graphs have the same set of signless Laplacian eigenvalues than have that of adjacency eigenvalues and that of Laplacian eigenvalues. With this strong basis, it is believed that studying graphs using signless Laplacian eigenvalues is more efficient than studying them by other eigenvalues associated with graphs [5]. In this note we focus on signless Laplacian matrices, so we call the eigenvalues of Q(G) the eigenvalues of G. Obviously Q(G) is symmetric, so has n real eigenvalues. Let q(G) denote the largest eigenvalue of Q(G). Note that q(G) ≥ Δ + 1 [9]. As given in [3, Corollary 3.9.3], a short proof of the previous lower bound of q(G) is by using interlacing property to show that q(G) ≥ q(K1,Δ ) = Δ + 1, where K1,Δ is a subgraph of G of order Δ + 1 and

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size Δ consisting of a vertex of maximum degree and Δ edges connecting the vertex to its D neighbors. By Perron–Frobenius Theorem, there exists a column vector x = n (x1 , x2 , . . . , xn )T with xi > 0 and i=1 x2i = 1 such that Q(G)x = q(G)x. The vector x is referred to as Perron eigenvector of Q(G). Let Gc denote the complement of G, i.e. the graph with the same vertex set V and with adjacency matrix A(Gc ) = J − I − A(G), where I is the identity matrix and J is the all ones matrix. Note that Q(G) + Q(Gc ) = (n − 2)I + J. By Cauchy–Schwarz inequality  T

x Jx =

n 

2 ≤n

xi

i=1

n 

x2i = nxT x.

i=1

Hence xT Q(Gc )x = xT [(n − 2)I + J − Q(G)] x ≤ 2n − 2 − q(G).

(2.1)

On the other hand, by direct matrix multiplication, 

xT Q(Gc )x =

(xi + xj )2 .

(2.2)

i
3. Main result The following is our main theorem. Theorem 3.1. Let G = (V, E) be a simple graph of order n with maximum degree Δ. Let λ (resp. μ) denote the maximum number of common neighbors of a pair of adjacent vertices (resp. nonadjacent vertices) of G. Let q(G) denote the largest eigenvalue of G. Then μ q(G) ≤ Δ − + 4



Δ−

μ 2 + (1 + λ)Δ + μ(n − 1) − Δ2 , 4

(3.1)

with equality if and only if G is a strongly regular graph with parameters (n, Δ, λ, μ). Proof. Since the graph G is fixed, we will omit the G in the symbols Q(G), q(G), A(G) and D(G). Let x be the Perron eigenvector of Q. Then (qI − D)x2 = (Q − D)x2 = Ax2 .

(3.2)

One the other hand (qI − D)x2 =

 i∈V

(q − di )2 x2i ≥ (q − Δ)2

(3.3)

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by using q ≥ Δ + 1, and also Ax2 = x A2 x =



di x2i + 2

i∈V

 (A2 )jk xj xk .

(3.4)

j
By using di ≤ Δ, and the definition of λ and μ to (3.4), 

Ax2 ≤ Δ + 2λ

xj xk + 2μ



xj xk .

(3.5)

j
j
Note that 

2λ

xj xk = λxT Ax ≤ λρ(A) ≤ λΔ,

(3.6)

j
and by (2.2) and (2.1), 2μ

 j
xj xk ≤

μ 2



(xj + xk )2 =

j
μ(2n − 2 − q) μxT Q(Gc )x ≤ . 2 2

(3.7)

Combining (3.3) and (3.5)–(3.7), one immediately has

 μ μ(2n − 2 − q) q +Δ2 −(1+λ)Δ−μ(n−1) = (q −Δ)2 − Δ + λΔ + ≤ 0. q 2 − 2Δ − 2 2 The inequality in (3.1) follows by solving the above quadratic polynomial in q. As xi > 0, the equality holds in (3.1) if and only if the equalities in (3.3) and (3.5) hold. This is equivalent to that G is strongly regular with parameters (n, Δ, λ, μ). 2 Remark 3.2. For upper bounds of the largest eigenvalue ρ(G) of A(G), it was shown in [14] that λ−μ + ρ(G) ≤ 2



λ−μ 2

2 + Δ + μ(n − 1),

with equality if and only if G is a strongly regular graph with parameters (n, Δ, λ, μ). Applying Courant–Weyl inequality q(G) ≤ Δ + ρ(G) [3, Theorem 2.8.1] and using the above upper bound of ρ(G), we have another upper bound of q(G) in a different form: λ−μ + q(G) ≤ Δ + 2



λ−μ 2

2 + Δ + μ(n − 1).

(3.8)

Inequalities (3.1) and (3.8) are incomparable, but in the case of equalities, both give characterizations of strongly regular graphs. An upper bound of the largest eigenvalue

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of L(G) is given in [11]. The graphs that obtain this upper bound include the class of strongly regular graphs and also other graphs. Other upper bounds of the largest eigenvalue of Q(G) or L(G) can be found in [4,6–8,10]. The extremal graphs of these upper bounds are other graphs, not related to strongly regular graphs. Acknowledgements The first author is honored to be an exchange student from HIT to NCTU. Both authors thank the anonymous referees for giving many valuable suggestions in the presentation of the paper. This research is supported by the Ministry of Science and Technology, Taiwan under the project NSC 102-2115-M-009-009-MY3. References [1] E. Bannai, T. Ito, Algebraic Combinatorics: Association Schemes, Benjamin–Cummings Lecture Note, vol. 58, The Benjamin/Cummings Publishing Company, Inc., London, 1984. [2] A. Brouwer, A. Cohen, A. Neumaier, Distance-Regular Graphs, Springer-Verlag, New York, 1989. [3] A.E. Brouwer, W.H. Haemers, Spectra of Graphs, Springer-Verlag, New York, 2012. [4] Y.H. Chen, R.Y. Pan, X.D. Zhang, Two sharp upper bounds for the signless Laplacian spectral radius of graphs, Discrete Math. Algorithms Appl. 3 (2) (2011) 185–191. [5] D. Cvetković, S.K. Simić, Towards a spectral theory of graphs based on the signless Laplacian, I, Publ. Inst. Math. (N.S.) 85 (99) (2009) 19–33. [6] K.C. Das, The Laplacian spectrum of a graph, Comput. Math. Appl. 48 (2004) 715–724. [7] K.C. Das, On conjectures involving second largest signless Laplacian eigenvalue of graphs, Linear Algebra Appl. 432 (2010) 3018–3029. [8] C. Elphick, C.A. Liu, A (forgotten) bound for the spectral radius of a graph, Taiwanese J. Math. 19 (2015) 1577–1829. [9] R. Grone, R. Merris, The Laplacian spectrum of a graph II, SIAM J. Discrete Math. 7 (1994) 221–229. [10] M. Liu, B. Liu, The signless Laplacian spread, Linear Algebra Appl. 432 (2010) 505–514. [11] F.-H. Lin, C.-W. Weng, A bound on the Laplacian spread which is tight for strongly regular graphs, Linear Algebra Appl. 494 (2016) 11–22. [12] J.H. van Lint, R.M. Wilson, A Course in Combinatorics, Cambridge University Press, 2001. [13] H. Minc, Nonnegative Matrices, John Wiley and Sons Inc., New York, 1988. [14] L. Shi, Z. Song, Upper bounds on the spectral radius of book-free and/or K2, -free graphs, Linear Algebra Appl. 420 (2007) 526–529.