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

Let ? _n i] be the ring of Gaussian integers modulo n. We construct for ?ni] a cubic mapping graph Γ(n) whose vertex set is all the elements of ?ni] and for which there is a directed edge from a ∈ ?ni] to b ∈ ?ni] if b = a ³. 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 ?ni] while Γ₂(n) is induced by all the zero-divisors of ?ni]. In addition, formulas on the heights of components and vertices in Γ(n) are presented.