On adjacency and (signless) Laplacian spectra of centralizer and co-centralizer graphs of some finite non-abelian groups

TL;DR

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

Contribution

It investigates the spectra of centralizer and co-centralizer graphs for certain finite non-abelian groups and establishes conditions for their integrality.

Findings

Determined the adjacency spectra of these graphs.

Established conditions for Laplacian spectral integrality.

Identified classes of groups with integral spectra.

Abstract

Let $G$ be a finite non abelian group. The centralizer graph of $G$ is a simple undirected graph $\Gamma_{cent}(G)$, whose vertices are the proper centralizers of $G$ and two vertices are adjacent if and only if their cardinalities are identical {\rm\cite{omer}}. The complement of the centralizer graph is called the co-centralizer graph. In this paper, we investigate the adjacency and (signless) Laplacian spectra of centralizer and co-centralizer graphs of some classes of finite non-abelian groups and obtain some conditions on a group so that the centralizer and co-centralizer graphs are adjacency, (signless) Laplacian integral.