Signless Laplacian Characteristic Polynomials of Complete Multipartite Graphs  

作　　者：LU Shi-fang ZHAO Hai-xing 

机构地区：[1]Department of Basic,Qinghai University,Xining 810016,China [2]Department of Mathematics,Qinghai Normal University,Xining 810008,China

出　　处：《Chinese Quarterly Journal of Mathematics》2012年第1期36-40,共5页数学季刊（英文版）

基　　金：Supported by the NSFC(60863006);Supported by the NCET(-06-0912);Supported by the Science-Technology Foundation for Middle-aged and Yong Scientist of Qinghai University(2011-QGY-8)

摘　　要：For a simple graph G,let matrix Q(G)=D(G) + A(G) be it's signless Laplacian matrix and Q G (λ)=det(λI Q) it's signless Laplacian characteristic polynomial,where D(G) denotes the diagonal matrix of vertex degrees of G,A(G) denotes its adjacency matrix of G.If all eigenvalues of Q G (λ) are integral,then the graph G is called Q-integral.In this paper,we obtain that the signless Laplacian characteristic polynomials of the complete multi-partite graphs G=K(n_1,n_2,···,n_t).We prove that the complete t-partite graphs K(n,n,···,n)t are Q-integral and give a necessary and sufficient condition for the complete multipartite graphs K(m,···,m)s(n,···,n)t to be Q-integral.We also obtain that the signless Laplacian characteristic polynomials of the complete multipartite graphs K(m,···,m,)s1(n,···,n,)s2(l,···,l)s3.For a simple graph G,let matrix Q（G）=D（G） ＋ A（G） be it’s signless Laplacian matrix and Q G （λ）=det（λI Q） it’s signless Laplacian characteristic polynomial,where D（G） denotes the diagonal matrix of vertex degrees of G,A（G） denotes its adjacency matrix of G.If all eigenvalues of Q G （λ） are integral,then the graph G is called Q-integral.In this paper,we obtain that the signless Laplacian characteristic polynomials of the complete multi-partite graphs G=K（n1,n2,···,nt）.We prove that the complete t-partite graphs K（n,n,···,n）t are Q-integral and give a necessary and sufficient condition for the complete multipartite graphs K（m,···,m）s（n,···,n）t to be Q-integral.We also obtain that the signless Laplacian characteristic polynomials of the complete multipartite graphs K（m,···,m,）s1（n,···,n,）s2（l,···,l）s3.

关 键 词：the signless Laplacian spectrum the complete multipartite graphs the Qintegral 

分 类 号：O157.5[理学—数学]