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

Title:

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

Creator:

Wei, Yangjiang, Nan, Jizhu, and Tang, Gaohua

Identifier:

https://cdk.lib.cas.cz/client/handle/uuid:f2d36387-de65-4438-a348-302ab66e30ad
uuid:f2d36387-de65-4438-a348-302ab66e30ad

Subject:

cubic mapping graph, cycle, and height

Type:

model:article and TEXT

Format:

bez média and svazek

Description:

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.

Language:

English

Rights:

http://creativecommons.org/publicdomain/mark/1.0/
policy:public

Coverage:

527-539

Source:

Czechoslovak Mathematical Journal | 2012 Volume:62 | Number:2

Harvested from:

CDK

Metadata only:

false