Finding cycles in the $k$-th power digraphs over the integers modulo a prime

Abstract

For $p$ prime and $k\ge 2$, let us define $G_p^{\left(k\right)}$ to be the digraph whose set of vertices is $\left\{0,1,2,\dots ,p-1\right\}$ such that there is a directed edge from a vertex $a$ to a vertex $b$ if $a^k\equiv bmodp$. We find a new way to decide if there is a cycle of a given length in a given graph $G_p^{\left(k\right)}$.