Signless Laplacian energies of non-commuting graphs of finite groups and related results

TL;DR

This paper computes the Signless Laplacian spectrum and energy of non-commuting graphs of finite groups, exploring their spectral properties, conditions for Q-integrality, and energetic hyper- and hypo-properties.

Contribution

It provides new spectral data and conditions for non-commuting graphs of finite groups, linking energy concepts with graph algebraic properties.

Findings

Computed Signless Laplacian spectrum and energy for various finite groups

Identified conditions for Q-integrality of non-commuting graphs

Analyzed hyper- and hypo-energetic properties of these graphs

Abstract

The non-commuting graph of a non-abelian group $G$ with center $Z(G)$ is a simple undirected graph whose vertex set is $G\setminus Z(G)$ and two vertices $x, y$ are adjacent if $xy \ne yx$. In this study, we compute Signless Laplacian spectrum and Signless Laplacian energy of non-commuting graphs of finite groups. We obtain several conditions such that the non-commuting graph of $G$ is Q-integral and observe relations between energy, Signless Laplacian energy and Laplacian energy. In addition, we look into the energetic hyper- and hypo-properties of non-commuting graphs of finite groups. We also assess whether the same graphs are Q-hyperenergetic and L-hyperenergetic.