Various spectral aspects of NCCC-graphs of certain finite non-abelian groups

TL;DR

This paper investigates the spectral properties and energies of non-commuting conjugacy class graphs of certain finite non-abelian groups, providing new insights into their structural and spectral characteristics.

Contribution

It computes various spectra and energies of NCCC-graphs for specific non-abelian groups and analyzes their integrality and energetic properties, offering novel spectral results.

Findings

Determined spectra and energies of NCCC-graphs for certain groups.

Identified conditions for graphs to be integral, L-integral, and Q-integral.

Compared different energies and classified graphs as borderenergetic or hyperenergetic.

Abstract

Let ${G}$ be a finite non-abelian group. The non-commuting conjugacy class graph (abbreviated as NCCC-graph) of $G$ is a simple undirected graph whose vertex set is the set of conjugacy classes of non-central elements of $G$ and two vertices $x^G$ and $y^G$ are adjacent to each other if $x'$ and $y'$ does not commute for all $x'\in x^G$ and $y'\in y^G$, where $x^G$ is the conjugacy class of $x \in G$. In this paper, we compute the spectrum, Laplacian spectrum, signless Laplacian spectrum and corresponding energies of NCCC-graphs of certain families of finite non-abelian groups. We determine whether these graphs are integral, L-integral and Q-integral. Further, we compare energy, Laplacian energy and signless Laplacian energy; and determine whether these graphs are borderenergetic, L-borderenergetic, Q-borderenergetic, hyperenergetic, L-hyperenergetic or Q-hyperenergetic.