The line graph of the crown graph is distance integral

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. A 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 $Cr(n)$. In this paper, by using the orbit partition method in algebraic graph theory, we determine the set of all distance eigenvalues of $L(Cr(n))$ and show that this graph is distance integral.