Previous |  Up |  Next

Article

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: 2016-04-07
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
.

Files

Files Size Format View
CzechMathJ_59-2009-1_3.pdf 235.6Kb application/pdf View/Open
Back to standard record
Partner of
EuDML logo