The $n$-total graph of an integral domain

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 $n\ge 1$ 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.