The triple zero graph of a commutative ring

Abstract

Let $R$ be a commutative ring with non-zero identity. We define the set of triple zero elements of $R$ by $TZ(R)=\{a\in Z(R)^{\ast}:$ there exists $b,c\in R\backslash\{0\}$ such that $abc=0$, $ab\neq0$, $ac\neq0$, $bc\neq0\}.$ In this paper, we introduce and study some properties of the triple zero graph of $R$ which is an undirected graph $TZ\Gamma(R)$ with vertices $TZ(R),$ and two vertices $a$ and $b$ are adjacent if and only if $ab\neq0$ and there exists a non-zero element $c$ of $R$ such that $ac\neq0$, $bc\neq0$, and $abc=0$. We investigate some properties of the triple zero graph of a general ZPI-ring $R,$ we prove that $diam(TZ\Gamma(R))\in\{0,1,2\}$ and $gr(G)\in\{3,\infty\}$.