The n-total graph of a commutative ring

TL;DR

This paper introduces the n-total graph of a commutative ring, exploring its properties and connections to ring structure, generalizing the total graph concept for rings.

Contribution

It defines the n-total graph for commutative rings and investigates its properties and implications for ring theory, extending previous total graph concepts.

Findings

Characterization of n-total graph properties

Connections between graph structure and ring properties

Generalization of total graph for n > 1

Abstract

Let $R$ be a commutative ring with $1\not = 0$, $Z(R)$ be the set of all zero-divisors of $R$, and $n \geq 1$. This paper introduces the $n$-total graph of a commutative ring $R$. The $n$-total graph of a commutative ring $R$, denoted by $n-T(R)$, is an undirected simple graph with vertex set $R$, such that two vertices $x, y$ in $R$ are connected by an edge if $x^n + y^n$ in $Z(R)$. Note that if $n =1$, then the $1$-total graph of $R$ is the total graph of $R$ in the sense of Anderson-Badawi's paper on the total graph of a commutative ring. In this paper, we study some graph properties and theoretical ring structure.