The structure of digraphs associated with the congruence $x^k\equiv y \hspace{4.44443pt}(\@mod \; n)$

Lawrence Somer; Michal Křížek

Displaying similar documents to “The structure of digraphs associated with the congruence $x^{k} \equiv y (mod n)$ ”

On the heights of power digraphs modulo $n$

Uzma Ahmad, Husnine Syed (2012)

Czechoslovak Mathematical Journal

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A power digraph, denoted by $G (n, k)$ , is a directed graph with $ℤ_{n} = {0, 1, \dots, n - 1}$ as the set of vertices and $E = {(a, b) : a^{k} \equiv b (mod n)}$ as the edge set. In this paper we extend the work done by Lawrence Somer and Michal Křížek: On a connection of number theory with graph theory, Czech. Math. J. 54 (2004), 465–485, and Lawrence Somer and Michal Křížek: Structure of digraphs associated with quadratic congruences with composite moduli, Discrete Math. 306 (2006), 2174–2185. The heights of the vertices and the components of $G (n, k)$ for $n \geq 1$ and...

Isomorphic digraphs from powers modulo $p$

Guixin Deng, Pingzhi Yuan (2011)

Czechoslovak Mathematical Journal

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Let $p$ be a prime. We assign to each positive number $k$ a digraph $G_{p}^{k}$ whose set of vertices is ${1, 2, ..., p - 1}$ and there exists a directed edge from a vertex $a$ to a vertex $b$ if $a^{k} \equiv b (mod p)$ . In this paper we obtain a necessary and sufficient condition for $G_{p}^{k_{1}} ≃ G_{p}^{k_{2}}$ .

Characterization of power digraphs modulo $n$

Uzma Ahmad, Syed Husnine (2011)

Commentationes Mathematicae Universitatis Carolinae

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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 $E = {(a, b) : a^{k} \equiv b (mod n)}$ 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.

Displaying similar documents to “The structure of digraphs associated with the congruence x k ≡ y ( mod n ) ”

On the heights of power digraphs modulo n

Isomorphic digraphs from powers modulo p

Characterization of power digraphs modulo n

Displaying similar documents to “The structure of digraphs associated with the congruence $x^{k} \equiv y (mod n)$ ”

On the heights of power digraphs modulo $n$

Isomorphic digraphs from powers modulo $p$

Characterization of power digraphs modulo $n$