Structure of cubic mapping graphs for the ring of Gaussian integers modulo $n$

Wei, Yangjiang ; Nan, Jizhu ; Tang, Gaohua

Czechoslovak Mathematical Journal, Tome 62 (2012) no. 2, pp. 527-539.

Résumé

Let ${\mathbb{Z}}_n[\mathrm{i}]$ be the ring of Gaussian integers modulo $n$. We construct for ${\mathbb{Z}}_n[\mathrm{i}]$ a cubic mapping graph $\mathrm{\Gamma }(n)$ whose vertex set is all the elements of\/ ${\mathbb{Z}}_n[\mathrm{i}]$ and for which there is a directed edge from $a\in {\mathbb{Z}}_n[\mathrm{i}]$ to $b\in {\mathbb{Z}}_n[\mathrm{i}]$ if $b=a^3$. This article investigates in detail the structure of $\mathrm{\Gamma }(n)$. We give suffcient and necessary conditions for the existence of cycles with length $t$. The number of $t$-cycles in ${\mathrm{\Gamma }}_1(n)$ is obtained and we also examine when a vertex lies on a $t$-cycle of ${\mathrm{\Gamma }}_2(n)$, where ${\mathrm{\Gamma }}_1(n)$ is induced by all the units of ${\mathbb{Z}}_n[\mathrm{i}]$ while ${\mathrm{\Gamma }}_2(n)$ is induced by all the zero-divisors of ${\mathbb{Z}}_n[\mathrm{i}]$. In addition, formulas on the heights of components and vertices in $\mathrm{\Gamma }(n)$ are presented.

Détail

DOI : 10.1007/s10587-012-0027-4

Classification : 05C05, 11A07, 13M05
Mots-clés : cubic mapping graph; cycle; height