The line graph of the crown graph is distance integral

TL;DR

This paper proves that the line graph of the crown graph has all integer distance eigenvalues, establishing it as distance integral using algebraic graph theory methods.

Contribution

It determines the complete set of distance eigenvalues for the line graph of crown graphs and proves their integrality, a novel result in spectral graph theory.

Findings

The line graph of the crown graph is distance integral.

All distance eigenvalues of L(Cr(n)) are integers.

The orbit partition method effectively analyzes the distance spectrum.

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.