Spectrum and genus of commuting graphs of some classes of finite rings

TL;DR

This paper investigates the spectral properties and genus of commuting graphs derived from certain finite rings, providing insights into their structural characteristics.

Contribution

It computes the spectrum and genus of commuting graphs for specific classes of finite rings, advancing understanding of their algebraic and topological features.

Findings

Spectral properties of commuting graphs are characterized.

Genus of commuting graphs is determined for certain finite rings.

Results contribute to the classification of finite rings via graph invariants.

Abstract

The commuting graph of a non-commutative ring $R$ with center $Z(R)$ is a simple undirected graph whose vertex set is $R\setminus Z(R)$ and two vertices $x, y$ are adjacent if and only if $xy = yx$. In this paper, we compute the spectrum and genus of commuting graphs of some classes of finite rings.