Geodesic
Characterization of power digraphs modulo $n$
Commentationes Mathematicae Universitatis Carolinae, Tome 52 (2011) no. 3, pp. 359-367 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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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.
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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     author = {Ahmad, Uzma and Husnine, Syed},
     title = {Characterization of power digraphs modulo $n$},
     journal = {Commentationes Mathematicae Universitatis Carolinae},
     pages = {359--367},
     year = {2011},
     volume = {52},
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     zbl = {1249.11002},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/CMUC_2011_52_3_a3/}
}
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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/