# Article

Full entry | PDF   (0.2 MB)
Keywords:
cubic mapping graph; cycle; height
Summary:
Let $\mathbb {Z}_n{\rm [i]}$ be the ring of Gaussian integers modulo $n$. We construct for $\mathbb {Z}_n{\rm [i]}$ a cubic mapping graph $\Gamma (n)$ whose vertex set is all the elements of\/ $\mathbb {Z}_n{\rm [i]}$ and for which there is a directed edge from $a \in \mathbb {Z}_n{\rm [i]}$ to $b \in \mathbb {Z}_n{\rm [i]}$ if $b = a^3$. This article investigates in detail the structure of $\Gamma (n)$. We give suffcient and necessary conditions for the existence of cycles with length $t$. The number of $t$-cycles in $\Gamma _1(n)$ is obtained and we also examine when a vertex lies on a $t$-cycle of $\Gamma _2(n)$, where $\Gamma _1(n)$ is induced by all the units of $\mathbb {Z}_n{\rm [i]}$ while $\Gamma _2(n)$ is induced by all the zero-divisors of $\mathbb {Z}_n{\rm [i]}$. In addition, formulas on the heights of components and vertices in $\Gamma (n)$ are presented.
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. T.: The Euler $\varphi$-function in the Gaussian integers. Am. Math. Mon. 90 (1983), 518-528. DOI 10.2307/2322785 | MR 0717096
[3] Meemark, Y., Wiroonsri, N.: The quadratic digraph on polynomial rings over finite fields. Finite Fields Appl. 16 (2010), 334-346. MR 2678622
[4] Somer, L., Křížek, M.: Structure of digraphs associated with quadratic congruences with composite moduli. Discrete Math. 306 (2006), 2174-2185. DOI 10.1016/j.disc.2005.12.026 | MR 2255611
[5] Somer, L., Křížek, M.: On symmetric digraphs of the congruence $x^k \equiv y\pmod n$. Discrete Math. 309 (2009), 1999-2009. DOI 10.1016/j.disc.2008.04.009 | MR 2510326
[6] 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.
[7] Tang, G. H., Su, H. D., Yi, Z.: Structure of the unit group of $\mathbb{Z}_n[i]$. J. Guangxi Norm. Univ., Nat. Sci. 28 (2010), 38-41 Chinese.
[8] 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. MR 2732885
[9] Wei, Y. J., Nan, J. Z., Tang, G. H.: The cubic mapping graph for the ring of Gaussian integers modulo $n$. Czech. Math. J. 61 (2011), 1023-1036. DOI 10.1007/s10587-011-0045-7 | MR 2886254

Partner of