The distance spectra of the derangement graphs

TL;DR

This paper investigates the distance spectra of derangement graphs, proving they have diameter 2 and determining their extremal eigenvalues, thus advancing understanding of their spectral properties.

Contribution

It provides a constructive proof of the diameter and explicitly determines the distance spectra and extremal eigenvalues of derangement graphs.

Findings

Connected derangement graphs have diameter 2.

Distance spectra of derangement graphs are explicitly determined.

Extremal distance eigenvalues are identified.

Abstract

In this paper, we consider the distance spectra of the derangement graphs. First we give a constructive proof that the connected derangement graphs are of diameter 2. Then we obtain their distance spectra. In particular, we determine all their extremal distance eigenvalues.