| Title:
|
The cubic mapping graph 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:
|
61 |
| Issue:
|
4 |
| Year:
|
2011 |
| Pages:
|
1023-1036 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
The article studies the cubic mapping graph $\Gamma (n)$ of $\mathbb {Z}_n[{\rm i}]$, the ring of Gaussian integers modulo $n$. For each positive integer $n>1$, the number of fixed points and the in-degree of the elements $\overline 1$ and $\overline 0$ in $\Gamma (n)$ are found. Moreover, complete characterizations in terms of $n$ are given in which $\Gamma _{2}(n)$ is semiregular, where $\Gamma _{2}(n)$ is induced by all the zero-divisors of $\mathbb {Z}_n[{\rm i}]$. (English) |
| Keyword:
|
Gaussian integers modulo $n$ |
| Keyword:
|
cubic mapping graph |
| Keyword:
|
fixed point |
| Keyword:
|
semiregularity |
| MSC:
|
05C05 |
| MSC:
|
11A07 |
| MSC:
|
13M05 |
| idZBL:
|
Zbl 1249.05061 |
| idMR:
|
MR2886254 |
| DOI:
|
10.1007/s10587-011-0045-7 |
| . |
| Date available:
|
2011-12-16T15:44:17Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/141804 |
| . |
| Reference:
|
[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 |
| Reference:
|
[2] Cross, J.: The Euler $\phi$-function in the Gaussian integers.Am. Math. Mon. 90 (1983), 518-528. Zbl 0525.12001, MR 0717096, 10.2307/2322785 |
| Reference:
|
[3] Pan, C. D., Pan, C. B.: Elementary Number Theory (2nd edition).Beijing University Publishing Company Beijing (2005), Chinese. |
| Reference:
|
[4] Skowronek-Kaziów, J.: Properties of digraphs connected with some congruence relations.Czech. Math. J. 59 (2009), 39-49. MR 2486614, 10.1007/s10587-009-0003-9 |
| Reference:
|
[5] Skowronek-Kaziów, J.: Some digraphs arising from number theory and remarks on the zero-divisor graph of the ring $\mathbb Z_n$.Inf. Process. Lett. 108 (2008), 165-169. MR 2452147, 10.1016/j.ipl.2008.05.002 |
| Reference:
|
[6] Somer, L., Křížek, M.: On a connection of number theory with graph theory.Czech. Math. J. 54 (2004), 465-485. MR 2059267, 10.1023/B:CMAJ.0000042385.93571.58 |
| Reference:
|
[7] 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. |
| Reference:
|
[8] Tang, G. H., Su, H. D., Yi, Z.: The structure of the unit group of $\mathbb Z_n[i]$.J. Guangxi Norm. Univ., Nat. Sci. 28 (2010), 38-41. |
| Reference:
|
[9] 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 |
| . |
