A typical graph structure of a ring

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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 Z_N(R)*, and two vertices x and y are adjacent if and only if xy is nilpotent and xy is nonzero, where Z_N(R)={x in R: xy 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 $Gamma_N^*(R)$.