Remarks on D-integral complete multipartite graphs

Abstract

A graph is called distance integral (or D-integral) if all eigenvalues of its distance matrix are integers. In their study of D-integral complete multipartite graphs, Yang and Wang (2015) posed two questions on the existence of such graphs. We resolve these questions and present some further results on D-integral complete multipartite graphs. We give the first known distance integral complete multipartite graphs $${K_{{p_1},{p_2},{p_3}}}$$ with p 1 < p 2 < p 3, and $${K_{{p_1},{p_2},{p_3},{p_4}}}$$ with p 1 < p 2 < p 3 < p 4, as well as the infinite classes of distance integral complete multipartite graphs $${K_{{a_1}{p_1},{a_2}{p_2},...,{a_s}{p_s}}}$$ with s = 5, 6.