On the Total Graph of a Finite Commutative Ring

TL;DR

This paper investigates the properties of the total graph of a finite commutative ring, including its structure, Eulerian nature, isomorphisms, and domination number, contributing to algebraic graph theory.

Contribution

It provides new characterizations of the total graph of finite commutative rings and explores conditions for Eulerian properties and isomorphisms with Cayley graphs.

Findings

Determined basic graph-theoretical properties of the total graph.

Identified conditions for the total graph to be Eulerian.

Established when the total graph is isomorphic to a Cayley graph.

Abstract

Let $R$ be a finite commutative ring with $1\ne 0$. In this article, we study the total graph of $R$, denoted by $\tau (R)$, determine some of its basic graph-theoretical properties, determine when it is Eulerian, and find some conditions under which this graph is isomorphic to $Cay(R,Z(R)\backslash\lbrace 0\rbrace)$. We shall also compute the domination number of $\tau (R).