Rings whose total graphs have genus at most one

TL;DR

This paper characterizes finite commutative rings whose total graphs are planar or toroidal, showing only finitely many such rings exist for each fixed genus and classifying those with genus at most one.

Contribution

It determines all finite commutative rings with total graphs of genus at most one and proves finiteness results for rings with a given total graph genus.

Findings

Classified all finite rings with total graphs of genus 0 or 1.

Proved only finitely many finite rings have total graph genus g for any positive integer g.

Established properties of the total graph related to the ring's structure.

Abstract

Let $R$ be a commutative ring with $\Z(R)$ its set of zero-divisors. In this paper, we study the total graph of $R$, denoted by $\T(\Gamma(R))$. It is the (undirected) graph with all elements of $R$ as vertices, and for distinct $x, y\in R$, the vertices $x$ and $y$ are adjacent if and only if $x + y\in\Z(R)$. We investigate properties of the total graph of $R$ and determine all isomorphism classes of finite commutative rings whose total graph has genus at most one (i.e., a planar or toroidal graph). In addition, it is shown that, given a positive integer $g$, there are only finitely many finite rings whose total graph has genus $g$.