Geodesic
Characterization of power digraphs modulo $n$
Commentationes Mathematicae Universitatis Carolinae, Tome 52 (2011) no. 3, pp. 359-367.

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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.
Classification : 05C20, 11A07, 11A15, 11A51, 20K01
Keywords: iteration digraph; isolated fixed points; Charmichael lambda function; Fermat numbers; Regular digraphs 
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     title = {Characterization of power digraphs modulo $n$},
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Ahmad, Uzma; Husnine, Syed. Characterization of power digraphs modulo $n$. Commentationes Mathematicae Universitatis Carolinae, Tome 52 (2011) no. 3, pp. 359-367. http://geodesic.mathdoc.fr/item/CMUC_2011__52_3_a3/