Abstract: Let Z[i]be the ring of Gaussian integers, γ∈Z[i]. Let 〈γ〉 denote the ideal of Z[i] generated by γ. The cubic mapping graph G(γ) over the quotient ring Z[i]/〈γ〉 is a digraph,where the vertices of G(γ) are the elements of Z[i]/〈γ〉, and there is a directed edge from α to β if β=α³. In this paper, the structure of G(γ) is investigated. The numbers of the fixed points and the in-degree of the vertices 0 and 1 are obtained. Moreover, the semiregularity of the graph G(γ) is characterized. Finally,the height in G(γ) of an arbitrary zero-divisor is determined.

Key words: Gaussian integers, cubic mapping graph, in-degree, cycles, semiregular