A power digraph, denoted by $G(n,k)$, is a directed graph with ${\mathbb{Z}}_n=\{0,1,\cdots ,n-1\}$ as the set of vertices and $E=\{(a,b):a^k\equiv b(modn)\}$ 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 $n\ge 1$ and...

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 $a^k\equiv b(modp)$. In this paper we obtain a necessary and sufficient condition for $G_p^{k_1}\simeq G_p^{k_2}$.

A power digraph modulo $n$, denoted by $G(n,k)$, is a directed graph with $Z_n=\{0,1,\cdots ,n-1\}$ as the set of vertices and $E=\{(a,b):a^k\equiv b(modn)\}$ 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.

We assign to each positive integer $n$ a digraph whose set of vertices is $H=\{0,1,\cdots ,n-1\}$ and for which there is a directed edge from $a\in H$ to $b\in H$ if $a^2\equiv b\left(modn\right)$. We establish necessary and sufficient conditions for the existence of isolated fixed points. We also examine when the digraph is semiregular. Moreover, we present simple conditions for the number of components and length of cycles. Two new necessary and sufficient conditions for the compositeness of Fermat numbers are also introduced.

Let ${\mathbb{Z}}_n\left[\mathrm{i}\right]$ be the ring of Gaussian integers modulo $n$. We construct for ${\mathbb{Z}}_n\left[\mathrm{i}\right]$ a cubic mapping graph $\Gamma \left(n\right)$ whose vertex set is all the elements of ${\mathbb{Z}}_n\left[\mathrm{i}\right]$ and for which there is a directed edge from $a\in {\mathbb{Z}}_n\left[\mathrm{i}\right]$ to $b\in {\mathbb{Z}}_n\left[\mathrm{i}\right]$ if $b=a^3$. This article investigates in detail the structure of $\Gamma \left(n\right)$. We give suffcient and necessary conditions for the existence of cycles with length $t$. The number of $t$-cycles in ${\Gamma }_1\left(n\right)$ is obtained and we also examine when a vertex lies on a $t$-cycle of ${\Gamma }_2\left(n\right)$, where ${\Gamma }_1\left(n\right)$ is induced by all the units of ${\mathbb{Z}}_n\left[\mathrm{i}\right]$ while ${\Gamma }_2\left(n\right)$ is induced...