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 $\mathbb {Z}_n{\rm [i]}$ 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.

Détail

DOI : 10.1007/s10587-012-0027-4

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