Distance Integral Complete Multipartite Graphs with s = 5; 6

YANG Ruo-song, WANG Li-gong

doi:10.13371/j.cnki.chin.q.j.m.2016.02.001

Chinese Quarterly Journal of Mathematics >

2016 , Vol. 31 >Issue 2: 111 - 117

DOI: https://doi.org/10.13371/j.cnki.chin.q.j.m.2016.02.001

Distance Integral Complete Multipartite Graphs with s = 5; 6

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School of Science,Northwestern Polytechnical University

YANG Ruo-song(1990-), male, native of Luoyang, Henan, an postgraduate student of Northwestern Polytechnical University, engages in graph theory and its applications; WANG Li-gong(corresponding author)(1968-), male, native of Xinzhou, Shanxi,a professor of Northwestern Polytechnical University, Ph.D.,engages in graph theory and its applications.

Received date: 2015-02-10

Online published: 2020-11-05

Supported by

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

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Abstract

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.

Key words： complete multipartite graph; distance matrix; distance integral; graph spectrum

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YANG Ruo-song, WANG Li-gong . Distance Integral Complete Multipartite Graphs with s = 5; 6[J]. Chinese Quarterly Journal of Mathematics, 2016 , 31(2) : 111 -117 . DOI: 10.13371/j.cnki.chin.q.j.m.2016.02.001

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