Czechoslovak Mathematical Journal, Vol. 61, No. 4, pp. 1023-1036, 2011

# The cubic mapping graph for the ring of Gaussian integers modulo $n$

## Yangjiang Wei, Jizhu Nan, Gaohua Tang

Y. J. Wei (corresponding author), J. Z. Nan, School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, P. R. China, e-mail: weiyangjiang2004@yahoo.com.cn, jznan@163.com; G. H. Tang, School of Mathematical Sciences, Guangxi Teachers Education University, Nanning 530023, P. R. China, e-mail: tanggaohua@163.com

Abstract: The article studies the cubic mapping graph $\Gamma(n)$ of $\mathbb{Z}_n[ i]$, 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 $\overline1$ and $\overline0$ 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 $\mathbb{Z}_n[ i]$.

Keywords: Gaussian integers modulo $n$, cubic mapping graph, fixed point, semiregularity

Classification (MSC 2010): 05C05, 11A07, 13M05

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