完全四部图的Seidel多项式及其谱(英文)

The Seidel polynomial and spectrum of the complete 4-partite graphs

设G是一个简单无向图,A(G)是图G的(0,1)邻接矩阵.定义S(G)=J-I-2A(G)是图G的Seidel矩阵,SG(λ)=det(λI-S(G))是图G的Seidel特征多项式(本文中简记为Seidel多项式),其中I是单位矩阵,J是全1矩阵.如果SG(λ)的特征值都是整数,则图G被称为是S-整图.本文主要研究完全四部图G=Kn1,n2,n3,n4的Seidel多项式及SG(λ)的特征根,给出了完全四部图Kn1,n2,n3,n4是S-整图的充要条件.

For a simple undirected graph G,let A（G） be the（0,1）-adjacency matrix of graph G,denote by the matrix S（G）=J-I-2A（G） the Seidel matrix,and SG（λ）=det（λI-S（G）） the Seidel characteristic polynomial of G（for simple the Seidel polynomial）,where I is identity matrix,J is a square matrix all of whose entries are equal to 1.If all eigenvalues of SG（λ） are integral,then the graph G be called S-integral.The Seidel polynomial and eigenvalues of SG（λ） are investigated for the complete 4-partite graphs G=Kn1,n2,n3,n4.The necessary and sufficient condition for the complete 4-partite graphs Kn1,n2,n3,n4 to be S-integral is given.