群U_(6n)上凯莱图的整性

Integrality of Cayley graphs over U_(6n)

令X是一个图,若X的邻接矩阵A(X)的所有特征值均为整数,则称图X为整图。本文研究了在群U_(6n)=〈a,b|a^(2n)=b3=1,a^(-1)ba=b^(-1)〉上凯莱图X(U6n,S)的整性,通过群表示理论中群的特征标和图的特征值之间的关系,刻画了在群U_(6n)上X(U_(6n),S)的谱,得到了X(U_(6n),S)是整图的充要条件。

A graph X is said to be integral if all eigenvalues of the adjacency matrix A(X)of X are integers.In this paper,the integrality of Cayley graphs X(U_(6n),S)over U_(6n)=〈a,b|a^(2n)=b 3=1,a^(-1)ba=b^(-1)〉are discussed,the spectra of X(U_(6n),S)over U_(6n)are characterized by the relationship between the eigenvalues of group and the characters of graph in the spectral group theory,and necessary and sufficient condition for X(U_(6n),S)to be the integral graphs are obtained.