Distance Integral Complete Multipartite Graphs with s=5, 6

作     者：YANG Ruo-song WANG Li-gong 

作者机构：School of Science Northwestern Polytechnical University Xi’an 710072 China 

出 版 物：《Chinese Quarterly Journal of Mathematics》 (数学季刊（英文版）)

年 卷 期：2016年第31卷第2期

页      面：111-117页

学科分类：07[理学] 070104[理学-应用数学] 0701[理学-数学] 

基　　金：Supported by the National Natural Science Foundation of China(11171273) Supported by the Graduate Starting Seed Fund of Northwestern Polytechnical University(Z2014173) 

主　　题：complete multipartite graph distance matrix distance integral graph spectrum 

摘      要：Let D(G) =(d_(ij))_(n×n) denote the distance matrix of a connected graph G with order n, where d_(ij) is equal to the distance between vertices viand vjin G. A graph is called distance integral if all eigenvalues of its distance matrix are integers. In 2014, Yang and Wang gave a sufficient and necessary condition for complete r-partite graphs K_(p1,p2,···,pr)=K_(a1·p1,a2·p2,···,as···ps) to be distance integral and obtained such distance integral graphs with s = 1, 2, 3, 4. However distance integral complete multipartite graphs K_(a1·p1,a2·p2,···,as·ps) with s 4 have not been found. In this paper, we find and construct some infinite classes of these distance integral graphs K_(a1·p1,a2·p2,···,as·ps) with s = 5, 6. The problem of the existence of such distance integral graphs K_(a1·p1,a2·p2,···,as·ps) with arbitrarily large number s remains open.