Some properties of the extended zero-divisor graph of the ring of Gaussian integers modulo n

Recently, Bennis and others studied an extension of the zero-divisor graph of a commutative ring R. They called this extension the extended zero-divisor graph of R, denoted by Γ(R). The graph Γ(R) has as set of vertices all the nonzero zero-divisors of R, Z(R)*, and two distinct vertices x and y are adjacent if there are nonnegative integers n and m such that xnym = 0 with xn ≠ 0 and ym ≠ 0. In this paper, we study several properties of the extended zero-divisor graph of the ring of Gaussian integers modulo n (Γ(Zn[i])). We characterize the positive integers n such that Γ(Zn[i]) = Γ(Zn[i]). The diameter and girth, as well as the positive integers n such that Γ(Zn[i]) is planar or outerplanar, are also determined