Geodesic
The cubic mapping graph for the ring of Gaussian integers modulo $n$
Czechoslovak Mathematical Journal, Tome 61 (2011) no. 4, pp. 1023-1036
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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 $\mathbb {Z}_n[{\rm i}]$.
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 $\mathbb {Z}_n[{\rm i}]$.
DOI : 10.1007/s10587-011-0045-7
Classification : 05C05, 11A07, 13M05
Keywords: Gaussian integers modulo $n$; cubic mapping graph; fixed point; semiregularity 
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     title = {The cubic mapping graph for the ring of {Gaussian} integers modulo $n$},
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Wei, Yangjiang; Nan, Jizhu; Tang, Gaohua. The cubic mapping graph for the ring of Gaussian integers modulo $n$. Czechoslovak Mathematical Journal, Tome 61 (2011) no. 4, pp. 1023-1036. doi: 10.1007/s10587-011-0045-7

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