On the signless laplacian spectral radius of some graphs

Mathematical Computation June 2014, Volume 3, Issue 2, PP.55-57

On the Signless Laplacian Spectral Radius of Some Graphs Jingming Zhang 1, 2, #, Tingzhu Huang 1 1. School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu, Sichuan, 611731, PR.China 2. College of Science, China University of Petroleum, Shandong, QingDao, 266580, PR. China #Email: zhangjm7519@126.com

Abstract The signless Laplacian spectral radius of a graph G is the largest eigenvalue of its signless Laplacian matrix. In this paper, we determine the largest signless Laplacian spectral among some graphs.

Keywords: Spectral Radius; Signless Laplcian Spectral Radius; Unicyclic Graph

1 INTRODUCTION Let G  (V (G), E(G)) be a simple connected graph with vertex set V  {v1 , v2 , , vn } and edge set E . Its adjacency matrix A(G)  (aij ) is defined as a n  n matrix (aij ) , where aij  1 it vi is adjacent to v j ; and aij  0 , otherwise.

Denote by d (vi ) or dG (vi ) the degree of the vertex vi (i  1, 2, , n) . Let Q(G)  D(G)  A(G) be the signless Laplacian matrix of a graph G , where D(G)  diag (d (v1 ), d (v2 ), , d (vn )) denotes the diagonal matrix of the vertex degrees of G . It is well known that A(G) is a real symmetric matrix and Q(G) is a positive semi-definite matrix.

Hence, the eigenvalues of can be ordered as q1 (G)  q2 (G)   qn (G)  0 . The largest eigenvalues of A(G) ( Q(G) , respectively) is called the spectral radius (signless Laplacian spectral radius, respectively) of G , denoted by  (G) ( q(G) , respectively). When G is connected, A(G) and Q(G) are nonnegative irreducible matrix. By the Perron-Frobenius Theorem, q(G) is simple and has a unique positive unit eigenvector. We refer to such the eigenvector corresponding to q(G) as the Perron vector of G .Two distinct edges in a graph G are independent if they are not adjacent in G . The characteristic polynomial of A(G) is det ( xI  A(G)) , which is denoted by (G) or (G, x) .The characteristic polynomial of Q(G) is det ( xI  Q(G)) , which is denoted by (G) or (G, x) . A unicyclic graph is a connected graph in which the number of vertices equals the number of edges. The investigation on the spectral radius (signless Laplacian spectral radius, respectively ) of graphs is an important topic in the theory of graph spectra, and some early results can go back to the very beginnings (see [1]). In this paper, we determine the largest signless Laplacian spectral radius among some graphs.

2 LEMMAS Lemma 2.1 ([2, 3]) Let G be a connected graph, and u, v be two vertices of G . Suppose that v1 , v2 , , vs  N (v) \ {N (u)  u} (1  s  d (v)) and x  ( x1 , x2 , , xn ) is the Perron vector of G , where xi corresponds to the vertex vi (1  i  n) . Let G * be the graph obtained from G by deleting the edges vvi and adding the edges uvi (1  i  s) . If xu  xv , then q(G)  q(G* ) .

Lemma 2.2 ([4, 5]) Let G be a graph on n vertices and m edges. Then (S (G))  xmn (G, x2 ) , where (G) and (G) are the characteristic polynomials of A(G) and Q(G) , respectively.

Lemma 2.3 ([6]) Let e  uv be an edge of G , and C (e) be the set of all circuits containing e . Then (G) satisfies (G)  (G  e)  (G  u  v)  2 (G  V (Z )), where the summation extends over all Z  C (e) .

Z

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From [7], we immediately have Corollary 2.4. Let (G) be the maximum degree of G . Then  (G)  (G) .

FIG.1

3 CONCLUSIONS Theorem 3.1 Let G1 ,

G2 and G3 be the graphs as Fig.1. Then for n  6 , q(G1 )  q(G2 ) , q(G1 )  q(G3 ) .

Proof. From Lemma 2.1, it is easy to see that q(G1 )  q(G2 ) . Now we prove that q(G1 )  q(G3 ) . From Lemma 2.3, we have ( S (G3 ))  ( S (G1 ))  x 2 ( x 2  1)n  2  ( x 4  3x 2  1)   4 [(n    3) x10  (20  6  6n) x8  (11n  49  10  ) x6  (54  3  6n) x 4  (n  25) x 2  4]

If

n  10

,

it

is

easy

to

prove

that

for

x  n   1 ,

x2  1  0

,

x4  3x2  1  0

and

(n    3) x  (20  6  6n) x  (11n  49  10 ) x  (54  6n  3 ) x  (n  25) x  4  0 . By Corollary 2.4, we 10

8

6

4

2

know that  (S (G1 ))  n    1 . Thus,  (S (G1 ))   (S (G3 )) (n  10) . For 6  n  9 , by direct calculation, we have  (S (G1 ))   (S (G3 )) . By Lemma 2.2, we have q(G1 )  q(G3 ) .

ACKNOWLEDGMENT This research is supported by NSFC (61170311), Chinese Universities Specialized Research Fund for the Doctoral Program (20110185110020), Sichuan Province Sci. \& Tech. Research Project (12ZC1802) and the Fundamental Research Funds for the Central Universities.

REFERENCES [1]

L. Collatz, U. Sinogowitz, Spektren endlicher grapen, Abh. Math. Sem. Univ. Hamburg 21(1957) 63-77.

[2]

D. Cvetkovi_c, P. Rowlinson, S. K. Simi_c, Signless Laplacian of _nite graphs, Linear Algebra Appl. 423(2007) 155-171.

[3]

Y. Hong, X. D. Zhang, Sharp upper and lower bounds for largest eigenvalue of the Laplacian matrices of trees, Discrete Mathematics, 296(2005) 187-197.

[4]

D. Cvetkovi_c, M. Doob, H. Sachs, Spectra of graphs, third ed., Johann Ambrosius Barth Verlag, Heidelberg, Leipzig, 1995.

[5]

B. Zhou, I. Gutman, A connection between ordinary and Laplacian spectra of bipartite graphs, Linear and Multilinear Algebra.

56(2008) 305-310.

[6]

A. J. Schwenk, R. J. Wilson, On the eigenvalues of a graph, in: L. W. Beineke, R. J. Wilson(Eds.), Selected Topics in Graph Theory, Academic Press, London, 1978, 307-336 (Chapter II).

[7]

Q. Li, K. Q. Feng, On the largest eigenvalue of graphs (Chinese), Acta Math. Appl. Sinica 2(1979) 167-175.

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AUTHORS 1Jing-Ming

Zhang was born in ShanDong province in

1975. PhD in School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu. The author’s major field is numerical algebra and scientific computation He once worked as a teacher in China University of Petroleum in ShanDong province. So far, he has published two papers in LAA and. nowadays; he is being interested in preconditioning techniques for large sparse linear systems.

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