# 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 .

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