On Simple Graphs Arising from Exponential Congruences

Abstract

We introduce and investigate a new class of graphs arrived from exponential congruences. For each pair of positive integers $a$ and $b$, let $G(n)$ denote the graph for which $V=\{0,1,\dots ,n-1\}$ is the set of vertices and there is an edge between $a$ and $b$ if the congruence $a^x\equiv b\hspace{0.17em}(\text{mod\hspace{0.17em}}n)$ is solvable. Let $n=p_1^{k_1}{p}_2^{k_2}\cdots p_r^{k_r}$ be the prime power factorization of an integer $n$, where are distinct primes. The number of nontrivial self-loops of the graph $G(n)$ has been determined and shown to be equal to ${\prod }_{i=1}^r(\varphi (p_i^{k_i})+1)$. It is shown that the graph $G(n)$ has $2^r$ components. Further, it is proved that the component ${\mathrm{\Gamma }}_p$ of the simple graph $G(p^2)$ is a tree with root at zero, and if $n$ is a Fermat's prime, then the component ${\mathrm{\Gamma }}_{\varphi (n)}$ of the simple graph $G(n)$ is complete.