| Title:
|
On semiregular digraphs of the congruence $x^k\equiv y \pmod n$ (English) |
| Author:
|
Somer, Lawrence |
| Author:
|
Křížek, Michal |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
48 |
| Issue:
|
1 |
| Year:
|
2007 |
| Pages:
|
41-58 |
| . |
| Category:
|
math |
| . |
| Summary:
|
We assign to each pair of positive integers $n$ and $k\geq 2$ a digraph $G(n,k)$ whose set of vertices is $H=\{0,1,\dots,n-1\}$ and for which there is a directed edge from $a\in H$ to $b\in H$ if $a^k\equiv b\pmod n$. The digraph $G(n,k)$ is semiregular if there exists a positive integer $d$ such that each vertex of the digraph has indegree $d$ or 0. Generalizing earlier results of the authors for the case in which $k=2$, we characterize all semiregular digraphs $G(n,k)$ when $k\geq 2$ is arbitrary. (English) |
| Keyword:
|
Chinese remainder theorem |
| Keyword:
|
congruence |
| Keyword:
|
group theory |
| Keyword:
|
dynamical system |
| Keyword:
|
regular and semiregular digraphs |
| MSC:
|
05C20 |
| MSC:
|
05C25 |
| MSC:
|
11A07 |
| MSC:
|
11A15 |
| MSC:
|
20K01 |
| idZBL:
|
Zbl 1174.05058 |
| idMR:
|
MR2338828 |
| . |
| Date available:
|
2009-05-05T17:01:12Z |
| Last updated:
|
2012-04-30 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/119637 |
| . |
| Reference:
|
[1] Křížek M., Luca F., Somer L.: 17 Lectures on Fermat Numbers: From Number Theory to Geometry.CMS Books in Mathematics, vol. 9, Springer New York (2001). Zbl 1010.11002, MR 1866957 |
| Reference:
|
[2] Lucheta C., Miller E., Reiter C.: Digraphs from powers modulo $p$.Fibonacci Quart. (1996), 34 226-239. Zbl 0855.05067, MR 1390409 |
| Reference:
|
[3] Niven I., Zuckerman H.S., Montgomery H.L.: An Introduction to the Theory of Numbers.fifth edition, John Wiley & Sons, New York (1991). Zbl 0742.11001, MR 1083765 |
| Reference:
|
[4] Somer L., Křížek M.: On a connection of number theory with graph theory.Czechoslovak Math. J. 54 (2004), 465-485. Zbl 1080.11004, MR 2059267 |
| Reference:
|
[5] Somer L., Křížek M.: Structure of digraphs associated with quadratic congruences with composite moduli.Discrete Math. 306 (2006), 2174-2185. Zbl 1161.05323, MR 2255611 |
| Reference:
|
[6] Wilson B.: Power digraphs modulo $n$.Fibonacci Quart. (1998), 36 229-239. Zbl 0936.05049, MR 1627384 |
| . |
