The distance spectrum of the line graph of the crown graph

Abstract

The distance eigenvalues of a connected graph $G$ are the eigenvalues of its distance matrix $D(G)$. A graph is called distance integral if all of its distance eigenvalues are integers. Let $n \geq 3$ be an integer. The crown graph $Cr(n)$ is a graph obtained from the complete bipartite graph $K_{n,n}$ by removing a perfect matching. Let $L(Cr(n))$ denote the line graph of the crown graph $Cr(n)$. Using the equitable partition method, the set of distinct distance eigenvalues of the graph $L(Cr(n))$ has been determined which shows that this graph is distance integral [S.Morteza Mirafzal, The line graph of the crown graph is distance integral, Linear and Multilinear Algebra 71, no. 4 (2023): 662-672]. The distance spectrum of the graph $L(Cr(n))$ has not been found yet. In this paper, having the set of distance eigenvalues of $L(Cr(n))$ in the hand, we determine the distance spectrum of this graph.