a typical graph structure of a ring

The zero-divisor graph of a commutative ring R with respect to nilpotent elements is asimple undirected graph Γ∗N (R) with vertex set ZN (R)∗, and two vertices x and y are adjacent if and only if xy is nilpotent and xy ̸= 0, where ZN (R) = {x ∈ R : xy is nilpotent, for some y ∈ R∗}. In this paper, we investigate the basic properties of Γ∗ (R). We discuss when it will be Eulerian and Hamiltonian. We further determine the genus of Γ∗ (R).