| Title:
|
Properties of digraphs connected with some congruence relations (English) |
| Author:
|
Skowronek-Kaziów, J. |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
59 |
| Issue:
|
1 |
| Year:
|
2009 |
| Pages:
|
39-49 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
The paper extends the results given by M. Křížek and L. Somer, {\it On a connection of number theory with graph theory}, Czech. Math. J. 54 (129) (2004), 465--485 (see [5]). For each positive integer $n$ define a digraph $\Gamma (n)$ whose set of vertices is the set $H=\{0,1,\dots ,n - 1\}$ and for which there is a directed edge from $a\in H$ to $b\in H$ if $a^3\equiv b\pmod n.$ The properties of such digraphs are considered. The necessary and the sufficient condition for the symmetry of a digraph $\Gamma (n)$ is proved. The formula for the number of fixed points of $\Gamma (n)$ is established. Moreover, some connection of the length of cycles with the Carmichael $\lambda $-function is presented. (English) |
| Keyword:
|
digraphs |
| Keyword:
|
Chinese remainder theorem |
| Keyword:
|
Carmichael $\lambda $-function |
| Keyword:
|
group theory |
| MSC:
|
05C20 |
| MSC:
|
05C25 |
| MSC:
|
11A15 |
| MSC:
|
20K01 |
| idZBL:
|
Zbl 1221.05183 |
| idMR:
|
MR2486614 |
| . |
| Date available:
|
2010-07-20T14:50:15Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/140462 |
| . |
| Reference:
|
[1] Bryant, S.: Groups, graphs and Fermat's last theorem.Amer. Math. Monthly 74 (1967), 152-156. Zbl 0163.02605, MR 0207824, 10.2307/2315605 |
| Reference:
|
[2] Carmichael, R. D.: Note on a new number theory function.Bull. Amer. Math. Soc. 16 (1910), 232-238 \JFM 41.0226.04. MR 1558896, 10.1090/S0002-9904-1910-01892-9 |
| Reference:
|
[3] Chassé, G.: Combinatorial cycles of a polynomial map over a commutative field.Discrete Math. 61 (1986), 21-26. MR 0850926, 10.1016/0012-365X(86)90024-5 |
| Reference:
|
[4] Harary, F.: Graph Theory.Addison-Wesley Publ. Company, London (1969). Zbl 0196.27202, MR 0256911 |
| Reference:
|
[5] Křížek, M., Somer, L.: 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:
|
[6] Křížek, M., Luca, F., Somer, L.: 17 Lectures on the Fermat Numbers. From Number Theory to Geometry.Springer-Verlag, New York (2001). MR 1866957 |
| Reference:
|
[7] Rogers, T. D.: The graph of the square mapping on the prime fields.Discrete Math. 148 (1996), 317-324. Zbl 0843.05048, MR 1368298, 10.1016/0012-365X(94)00250-M |
| Reference:
|
[8] Sierpiński, W.: Elementary Theory of Numbers.North-Holland (1988). MR 0930670 |
| Reference:
|
[9] Szalay, L.: A discrete iteration in number theory.BDTF Tud. Közl. 8 (1992), 71-91 Hungarian. Zbl 0801.11011 |
| . |
