Genus of commuting graphs of some classes of finite rings

TL;DR

This paper calculates the genus of commuting graphs for certain finite rings of specific orders and characterizes when these graphs are planar or toroidal, advancing understanding of their topological properties.

Contribution

It provides explicit genus calculations for commuting graphs of finite rings of orders $p^4$, $p^5$, $p^2q$, and $p^3q$, and characterizes planarity and toroidality.

Findings

Genus values for rings of orders $p^4$, $p^5$, $p^2q$, $p^3q$ are determined.

Conditions for commuting graphs to be planar or toroidal are characterized.

The topological classification of these graphs is advanced.

Abstract

In this paper, we compute the genus of commuting graphs of non-commutative rings of order $p^4$, $p^5$, $p^2q$ and $p^3q$, where $p$ and $q$ are prime integers. We also characterize those finite rings such that their commuting graphs are planar or toroidal.