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 $G_p^{k_1}\simeq G_p^{k_2}$.