| Title:
|
Characterization of power digraphs modulo $n$ (English) |
| Author:
|
Ahmad, Uzma |
| Author:
|
Husnine, Syed |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
52 |
| Issue:
|
3 |
| Year:
|
2011 |
| Pages:
|
359-367 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
A power digraph modulo $n$, denoted by $G(n,k)$, is a directed graph with $Z_{n}=\{0,1,\dots, n-1\}$ as the set of vertices and $E=\{(a,b): a^{k}\equiv b\pmod n\}$ as the edge set, where $n$ and $k$ are any positive integers. In this paper we find necessary and sufficient conditions on $n$ and $k$ such that the digraph $G(n,k)$ has at least one isolated fixed point. We also establish necessary and sufficient conditions on $n$ and $k$ such that the digraph $G(n,k)$ contains exactly two components. The primality of Fermat number is also discussed. (English) |
| Keyword:
|
iteration digraph |
| Keyword:
|
isolated fixed points |
| Keyword:
|
Charmichael lambda function |
| Keyword:
|
Fermat numbers |
| Keyword:
|
Regular digraphs |
| MSC:
|
05C20 |
| MSC:
|
11A07 |
| MSC:
|
11A15 |
| MSC:
|
11A51 |
| MSC:
|
20K01 |
| idZBL:
|
Zbl 1249.11002 |
| idMR:
|
MR2843229 |
| . |
| Date available:
|
2011-08-15T19:13:32Z |
| Last updated:
|
2013-10-14 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/141608 |
| . |
| Reference:
|
[1] Wilson B.: Power digraphs modulo $n$.Fibonacci Quart. 36 (1998), 229–239. Zbl 0936.05049, MR 1627384 |
| Reference:
|
[2] Lucheta C., Miller E., Reiter C.: Digraphs from powers modulo $p$.Fibonacc Quart. 34 (1996), 226–239. Zbl 0855.05067, MR 1390409 |
| Reference:
|
[3] Burton D.M.: Elementary Number Theory.McGraw-Hill, 2007. Zbl 1084.11001 |
| Reference:
|
[4] Chartrand G., Oellermann O.R.: Applied and Algorithmic Graph Theory.McGraw-Hill, New York, 1993. MR 1211413 |
| Reference:
|
[5] Kramer-Miller J.: Structural properties of power digraphs modulo $n$.in Proceedings of the 2009 Midstates Conference for Undergraduate Research in Computer Science and Mathematics, Oberlin, Ohio, 2009, pp. 40–49. |
| Reference:
|
[6] Ellson J., Gansner E., Koutsofios L., North S.C., Woodhull G.: Graphviz - open source graph drawing tools.version $2.26.3$, http://www.graphviz.org. Zbl 1054.68583 |
| Reference:
|
[7] Somer L., Křížek M.: On a connection of number theory with graph theory.Czechoslovak Math. J. 54 (2004), 465–485. MR 2059267, 10.1023/B:CMAJ.0000042385.93571.58 |
| Reference:
|
[8] Somer L., Křížek M.: Structure of digraphs associated with quadratic congruences with composite moduli.Discrete Math. 306 (2006), 2174–2185. MR 2255611, 10.1016/j.disc.2005.12.026 |
| Reference:
|
[9] Somer L., Křížek M.: On semi-regular digraphs of the congruence $x^{k}\equiv y \pmod n$.Comment. Math. Univ. Carolin. 48 (2007), no. 1, 41–58. MR 2338828 |
| Reference:
|
[10] Somer L., Křížek M.: On symmetric digraphs of the congruence $x^{k}\equiv y \pmod n$.Discrete Math. 309 (2009), 1999–2009. 10.1016/j.disc.2008.04.009 |
| Reference:
|
[11] Szalay L.: A discrete iteration in number theory.BDTF Tud. Közl. 8 (1992), 71–91. Zbl 0801.11011 |
| Reference:
|
[12] MATLAB,: The language of technical computing.version 7.0.0.19920 (R14). |
| Reference:
|
[13] Deo N.: Graph theory with Application to Engineering and Computer Sciences.Prentice-Hall of India private Limited, 1990. MR 0360322 |
| Reference:
|
[14] Carmichael R.D.: Note on a new number theory function.Bull. Amer. Math. Soc. 16 (1910), 232–238. MR 1558896, 10.1090/S0002-9904-1910-01892-9 |
| Reference:
|
[15] Husnine S.M., Ahmad U., Somer L.: On symmetries of power digraphs.Utilitas Mathematica(to appear). MR 2840802 |
| Reference:
|
[16] Skowronek-Kaziów J.: Properties of digraphs connected with some congruences relations.Czechoslovak Math. J. 59 (2009), 39–49. MR 2486614, 10.1007/s10587-009-0003-9 |
| . |
