| Title:
|
Structure of cubic mapping graphs for the ring of Gaussian integers modulo $n$ (English) |
| Author:
|
Wei, Yangjiang |
| Author:
|
Nan, Jizhu |
| Author:
|
Tang, Gaohua |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
62 |
| Issue:
|
2 |
| Year:
|
2012 |
| Pages:
|
527-539 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| 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. (English) |
| Keyword:
|
cubic mapping graph |
| Keyword:
|
cycle |
| Keyword:
|
height |
| MSC:
|
05C05 |
| MSC:
|
11A07 |
| MSC:
|
13M05 |
| idZBL:
|
Zbl 1261.05037 |
| idMR:
|
MR2990192 |
| DOI:
|
10.1007/s10587-012-0027-4 |
| . |
| Date available:
|
2012-06-08T09:51:17Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/142844 |
| . |
| Reference:
|
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| Reference:
|
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| Reference:
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| Reference:
|
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| Reference:
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| Reference:
|
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| Reference:
|
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| Reference:
|
[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 |
| Reference:
|
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| . |
