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Keywords:
distance spectrum; integral graph; distance integral graph; complete multipartite graph
distance spectrum; integral graph; distance integral graph; complete multipartite graph
Summary:
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 $K_{a_{1} p_{1},a_{2} p_{2},\ldots ,a_{s} p_{s}}$ with $s=5,6$.
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