The distance spectrum of the line graph of the crown graph

TL;DR

This paper determines the complete distance spectrum of the line graph of the crown graph, building on prior work that identified its distance eigenvalues using equitable partitions.

Contribution

It provides the explicit distance spectrum of the line graph of the crown graph, which was previously unknown, using the known set of distance eigenvalues.

Findings

The distance spectrum of L(Cr(n)) is explicitly determined.

L(Cr(n)) is shown to be distance integral.

The paper extends spectral graph theory results for crown graphs.

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.