Geodesic
Properties of digraphs connected with some congruence relations
Czechoslovak Mathematical Journal, Tome 59 (2009) no. 1, pp. 39-49.

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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.
Classification : 05C20, 05C25, 11A15, 20K01
Keywords: digraphs; Chinese remainder theorem; Carmichael $\lambda $-function; group theory 
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Skowronek-Kaziów, J. Properties of digraphs connected with some congruence relations. Czechoslovak Mathematical Journal, Tome 59 (2009) no. 1, pp. 39-49. http://geodesic.mathdoc.fr/item/CMJ_2009__59_1_a2/