A note on the distance spectra of co-centralizer graphs

TL;DR

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

Contribution

It investigates the distance and Laplacian spectra of co-centralizer graphs of finite non-abelian groups and provides conditions for spectral integrality.

Findings

Determined the spectra for specific classes of groups.

Established conditions for spectral integrality of co-centralizer graphs.

Analyzed the impact of group structure on graph spectra.

Abstract

Let $G$ be a finite non abelian group. The centralizer graph of $G$ is a simple undirected graph $\Gamma_{cent}(G)$, whose vertex set consists of proper centralizers of $G$ and two vertices are adjacent if and only if their cardinalities are identical [6]. We call the complement of the centralizer graph as the co-centralizer graph. In this paper, we investigate the distance, distance (signless) Laplacian spectra of co-centralizer graphs of some classes of finite non-abelian groups, and obtain some conditions on a group so that the co-centralizer graph is distance, distance (signless) Laplacian integral.