Distance integral complete multipartite graphs with $s=5,6$

Abstract

Let $D(G)=(d_{ij}{)}_{n\times n}$ denote the distance matrix of a connected graph $G$ with order $n$, where $d_{ij}$ is equal to the distance between vertices $v_i$ and $v_j$ in $G$. 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 $K_{p_1,p_2,\dots ,p_r}=K_{a_1\cdot p_1,a_2\cdot p_2,\dots ,a_s\cdot p_s}$ to be distance integral and obtained such distance integral graphs with $s=1,2,3,4$. However distance integral complete multipartite graphs $K_{a_1\cdot p_1,a_2\cdot p_2,\dots ,a_s\cdot p_s}$ with $s>4$ have not been found. In this paper, we find and construct some infinite classes of these distance integral graphs $K_{a_1\cdot p_1,a_2\cdot p_2,\dots ,a_s\cdot p_s}$ with $s=5,6$. The problem of the existence of such distance integral graphs $K_{a_1\cdot p_1,a_2\cdot p_2,\dots ,a_s\cdot p_s}$ with arbitrarily large number $s$ remains open.

arXiv comment6 pages, 1 table

References in corpus (1)

• On Distance Spectral Radius and Distance Energy of Graphs