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Chinese remainder theorem; congruence; group theory; dynamical system; regular and semiregular digraphs
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.
[1] Křížek M., Luca F., Somer L.: 17 Lectures on Fermat Numbers: From Number Theory to Geometry. CMS Books in Mathematics, vol. 9, Springer New York (2001). MR 1866957 | Zbl 1010.11002
[2] Lucheta C., Miller E., Reiter C.: Digraphs from powers modulo $p$. Fibonacci Quart. (1996), 34 226-239. MR 1390409 | Zbl 0855.05067
[3] Niven I., Zuckerman H.S., Montgomery H.L.: An Introduction to the Theory of Numbers. fifth edition, John Wiley & Sons, New York (1991). MR 1083765 | Zbl 0742.11001
[4] Somer L., Křížek M.: On a connection of number theory with graph theory. Czechoslovak Math. J. 54 (2004), 465-485. MR 2059267 | Zbl 1080.11004
[5] Somer L., Křížek M.: Structure of digraphs associated with quadratic congruences with composite moduli. Discrete Math. 306 (2006), 2174-2185. MR 2255611 | Zbl 1161.05323
[6] Wilson B.: Power digraphs modulo $n$. Fibonacci Quart. (1998), 36 229-239. MR 1627384 | Zbl 0936.05049
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