关于完全s部整图

F.Harary和A.J.Schwenk(Lecture Notes in Mathematics.Berlin:Springer-Verlag,1974,406:46-51.)提出了整图的概念,即当无向图G的邻接矩阵A的特征值都是整数时,G称为整图.目前,人们已经研究了n类简单整图的性质,并得到了一些有趣的结果.运用线性代数方法证明了两个结论:设r,r1,r2,s是正整数,那么:1)完全s部图K(r,r,…,r)是整图;2)完全2部图K(r1,r2)是整图的充要条件是r1r2为完全平方数.

On the Complete s-partite Graph

In 1974,F.Haraiy and A.J.Schwenk proposed the concept of integral graph.If all eigenvalues of the adjacent matrix A of a undirected graph G are integers,then G is called an integral graph.In recent years,some authors studied the properties of the n-kind of simple integral graphs,and obtained many interesting results.In this paper,by using some linear algebraic methods,we prove the following two conclusions: let r,r1,r2,s be positive integers,then 1) the complete s-partite graph K(r,r,…,r) is integral,2) the complete bipartite graph K(r1,r2) is integral if and only if r1r2 is a complete square number.