On the distance & distance (signless) Laplacian spectra of non-commuting graphs

TL;DR

This paper studies the spectral properties of non-commuting graphs of finite non-abelian groups, focusing on their distance and Laplacian spectra, and identifies conditions for these spectra to be integral.

Contribution

It analyzes the distance and signless Laplacian spectra of non-commuting graphs for certain groups and establishes criteria for spectral integrality.

Findings

Identifies conditions for distance spectrum integrality.

Determines when the signless Laplacian spectrum is integral.

Provides spectral characterizations for specific group classes.

Abstract

Let $Z(G)$ be the centre of a finite non-abelian group $G.$ The non-commuting graph of $G$ is a simple undirected graph with vertex set $G\setminus Z(G),$ and two vertices $u$ and $v$ are adjacent if and only if $uv\ne vu.$ In this paper, we investigate the distance, distance (signless) Laplacian spectra of non-commuting graphs of some classes of finite non-abelian groups, and obtain some conditions on a group so that the non-commuting graph is distance, distance (signless) Laplacian integral.