A typical graph structure of a ring

Abstract

The zero-divisor graph of a commutative ring $R$ with respect to nilpotent elements is a simple undirected graph $Gamma_N^*(R)$ with vertex set $mathcal{Z}_N(R)^*$, and two vertices $x$ and $y$ are adjacent if and only if $xy$ is nilpotent and $xyneq 0$, where $mathcal{Z}_N(R)={xin R: xy~text{is nilpotent, for some} yin R^*}$. In this paper, we investigate the basic properties of $Gamma_N^*(R)$. We discuss when it will be Eulerian and Hamiltonian. We further determine the genus of $Gamma_N^*(R)$.