A typical graph structure of a ring

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 xy\neq ۰, where \mathcal{Z}_N(R)=\{x\in R: xy~\text{is nilpotent, for some} y\in 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).

Keywords:

local ring , nilpotent , planar , Artinian ring