Structure of cubic mapping graphs for the ring of Gaussian integers modulo n

Let ℤn[i] be the ring of Gaussian integers modulo n. We construct for ℤn[i] a cubic mapping graph Γ(n) whose vertex set is all the elements of ℤn[i] and for which there is a directed edge from a ∈ ℤn[i] to b ∈ ℤn[i] if b = a3. This article investigates in detail the structure of Γ(n). We give suffcient and necessary conditions for the existence of cycles with length t. The number of t-cycles in Γ1(n) is obtained and we also examine when a vertex lies on a t-cycle of Γ2(n), where Γ1(n) is induced by all the units of ℤn[i] while Γ2(n) is induced by all the zero-divisors of ℤn[i]. In addition, formulas on the heights of components and vertices in Γ(n) are presented.