On the spectra of Commuting Graphs

TL;DR

This paper analyzes the spectral properties of commuting graphs of finite groups, linking group theory with graph spectra, and introduces a method to identify the group's center using spectral data.

Contribution

It provides a complete spectral characterization of commuting graphs and a novel spectral method to determine the group's center from the Laplacian spectrum.

Findings

Spectra of Laplacian, signless Laplacian, and adjacency matrices are fully described.

Graph invariants like diameter and clique number are determined from spectra.

Method to find the group's center using Laplacian spectrum is introduced.

Abstract

We describe the complete spectra of Laplacian, signless Laplacian, and adjacency matrices associated with the commuting graphs of a finite group using group theoretic information. We provide a method to find the center of a group by only using the Laplacian of the commuting graph of the group. The graph invariants (such as diameter, clique number, and mean distance) of the commuting graph associated with a finite group are determined. We produce a number of examples to illustrate the applications of our results.