# Article

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Keywords:
Gaussian integers modulo $n$; cubic mapping graph; fixed point; semiregularity
Summary:
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 $\mathbb {Z}_n[{\rm i}]$.
References:
[1] Osba, E. Abu, Henriksen, M., Alkam, O., Smith, F. A.: The maximal regular ideal of some commutative rings. Commentat. Math. Univ. Carol. 47 (2006), 1-10. MR 2223962
[2] Cross, J.: The Euler $\phi$-function in the Gaussian integers. Am. Math. Mon. 90 (1983), 518-528. DOI 10.2307/2322785 | MR 0717096 | Zbl 0525.12001
[3] Pan, C. D., Pan, C. B.: Elementary Number Theory (2nd edition). Beijing University Publishing Company Beijing (2005), Chinese.
[4] Skowronek-Kaziów, J.: Properties of digraphs connected with some congruence relations. Czech. Math. J. 59 (2009), 39-49. DOI 10.1007/s10587-009-0003-9 | MR 2486614
[5] Skowronek-Kaziów, J.: Some digraphs arising from number theory and remarks on the zero-divisor graph of the ring $\mathbb Z_n$. Inf. Process. Lett. 108 (2008), 165-169. DOI 10.1016/j.ipl.2008.05.002 | MR 2452147
[6] Somer, L., Křížek, M.: On a connection of number theory with graph theory. Czech. Math. J. 54 (2004), 465-485. DOI 10.1023/B:CMAJ.0000042385.93571.58 | MR 2059267
[7] Su, H. D., Tang, G. H.: The prime spectrum and zero-divisors of $\mathbb Z_n[i]$. J. Guangxi Teach. Edu. Univ. 23 (2006), 1-4.
[8] Tang, G. H., Su, H. D., Yi, Z.: The structure of the unit group of $\mathbb Z_n[i]$. J. Guangxi Norm. Univ., Nat. Sci. 28 (2010), 38-41.
[9] Wei, Y. J., Nan, J. Z., Tang, G. H., Su, H. D.: The cubic mapping graphs of the residue classes of integers. Ars Combin. 97 (2010), 101-110. MR 2732885

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