The distance spectrum of the line graph of the crown graph
S. Morteza Mirafzal
TL;DR
This work determines the distance spectrum of the line graph $L(Cr(n))$ of crown graphs $Cr(n)$. By leveraging the known distinct distance eigenvalues from equitable partitions and exploiting a commuting set of matrices $\{A,J,I,A_3\}$, the authors diagonalize the distance matrix and show that the distance eigenvalues are $\{(2n^2-4n+3)^1, 1^a, (-1)^b, (-n+3)^{n-1}, (-n-1)^{n-1}\}$ with explicit multiplicities $a=\tfrac{1}{2}(n^2-3n)$ and $b=\tfrac{1}{2}(n^2-3n+2)$. This yields a complete distance spectrum for $L(Cr(n))$, enriching the class of distance‑spectrum results for line graphs of distance‑transitive graphs and contributing to the study of distance‑integral graphs. The findings rely on the diameter‑3 structure and the interplay between the adjacency, all-ones, and distance‑3 matrices to obtain tight multiplicity formulas.
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.