The $n$-total graph of an integral domain

TL;DR

This paper introduces the $n$-total graph of a ring formed from an integral domain or product of domains, exploring its properties and the underlying ring structure based on the sum of powers belonging to a union of prime ideals.

Contribution

It defines the $n$-total graph for rings involving integral domains and prime ideals, and analyzes its properties and implications for ring structure.

Findings

Characterization of the $n$-total graph properties

Connections between graph structure and ring properties

Insights into the algebraic structure via graph analysis

Abstract

Let $R$ be a finite product of integral domains and $D$ be a union of prime ideals (it is possible that $R$ is just an integral domain). Let $n \geq 1$ be a positive integer. This paper introduces the $n$-total graph of a $(R, D)$. The $n$-total graph of $(R, D)$, 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 $x^n + y^n \in D$. In this paper, we study some graph properties and theoretical ring structure.