A Note on the Isomorphism Problem for Monomial Digraphs

Let $p$ be a prime $e$ be a positive integer, $q = p^e$, and let $\mathbb{F}_q$ denote the finite field of $q$ elements. Let $m,n$, $1\le m,n\le q-1$, be integers. The monomial digraph $D= D(q;m,n)$ is defined as follows: the vertex set of $D$ is $\mathbb{F}_q^2$, and $((x_1,x_2),(y_1,y_2))$ is an arc in $D$ if $x_2 + y_2 = x_1^m y_1^n$. In this note we study the question of isomorphism of monomial digraphs $D(q;m_1,n_1)$ and $D(q;m_2,n_2)$. Several necessary conditions and several sufficient conditions for the isomorphism are found. We conjecture that one simple sufficient condition is also a necessary one.