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Gaussian integers modulo $n$; cubic mapping graph; fixed point; semiregularity
The article studies the cubic mapping graph $\Gamma (n)$ of $\mathbb {Z}_n[{\rm i}]$, the ring of Gaussian integers modulo $n$. For each positive integer $n>1$, the number of fixed points and the in-degree of the elements $\overline 1$ and $\overline 0$ in $\Gamma (n)$ are found. Moreover, complete characterizations in terms of $n$ are given in which $\Gamma _{2}(n)$ is semiregular, where $\Gamma _{2}(n)$ is induced by all the zero-divisors of $\mathbb {Z}_n[{\rm i}]$.
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