Geodesic
On semiregular digraphs of the congruence $x^k\equiv y \pmod n$
Commentationes Mathematicae Universitatis Carolinae, Tome 48 (2007) no. 1, pp. 41-58.

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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.
Classification : 05C20, 05C25, 11A07, 11A15, 20K01
Keywords: Chinese remainder theorem; congruence; group theory; dynamical system; regular and semiregular digraphs 
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Somer, Lawrence; Křížek, Michal. On semiregular digraphs of the congruence $x^k\equiv y \pmod n$. Commentationes Mathematicae Universitatis Carolinae, Tome 48 (2007) no. 1, pp. 41-58. http://geodesic.mathdoc.fr/item/CMUC_2007__48_1_a3/