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).