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 $TZ(R)={ain Z(R)^{ast}:$ there exists $b,cin Rbackslash{0}$ such that $abc=0$, $abneq0$, $acneq0$, $bcneq0}.$ In this paper, we introduce and study some properties of the triple zero graph of $R$ which is an undirected graph $TZGamma(R)$ with vertices $TZ(R),$ and two vertices $a$ and $b$ are adjacent if and only if $abneq0$ and there exists a non-zero element $c$ of $R$ such that $acneq0$, $bcneq0$, and $abc=0$. We investigate some properties of the triple zero graph of a general ZPI-ring $R,$ we prove that $diam(TZGamma(R))in{0,1,2}$ and $gr(G)in{3,infty}$.