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

TL;DR

This paper constructs infinite classes of distance integral complete multipartite graphs with 5 and 6 parts, extending previous work that only covered up to 4 parts, and highlights the open problem for larger s.

Contribution

It introduces new infinite classes of distance integral complete multipartite graphs with s=5,6, expanding the known cases beyond s=4.

Findings

Constructed infinite classes of distance integral graphs with s=5.

Constructed infinite classes of distance integral graphs with s=6.

Identified open problem for graphs with arbitrarily large s.

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 $K_{p_{1},p_{2},\ldots,p_{r}}=K_{a_{1}\cdot p_{1},a_{2}\cdot p_{2},\ldots,a_{s}\cdot p_{s}}$ to be distance integral and obtained such distance integral graphs with $s=1,2,3,4$. However distance integral complete multipartite graphs $K_{a_{1}\cdot p_{1},a_{2}\cdot p_{2},\ldots,a_{s}\cdot p_{s}}$ with $s>4$ have not been found. In this paper, we find and construct some infinite classes of these distance integral graphs $K_{a_{1}\cdot p_{1},a_{2}\cdot p_{2},\ldots,a_{s}\cdot p_{s}}$ with $s=5,6$. The problem of the existence of such distance integral graphs $K_{a_{1}\cdot p_{1},a_{2}\cdot p_{2},\ldots,a_{s}\cdot p_{s}}$ with arbitrarily large number $s$ remains open.