Diameter of Some Monomial Digraphs

Let $p$ be a prime, $e$ a positive integer, $q=p^e$, and let ${\mathbb{F}}_q$ denote the finite field of $q$ elements. Let $f_i : $\mathbb{F}$_q^2\to$\mathbb{F}$_q$bearbitraryfunctions,where$1\le i\le l$,$i$and$l$areintegers\mathrm{.}Thedigraph$D = D(q;\bf{f})$,where${\bf f}=(f_1,\dotso,f_l) : $\mathbb{F}$_q^2\to$\mathbb{F}$_q^l$, is defined as follows. The vertex set of $D$ is ${\mathbb{F}}_q^{l+1}$. There is an arc from a vertex $\mathbf{x}=(x_1,\dots ,x_{l+1})$ to a vertex $\mathbf{y}=(y_1,\dots ,y_{l+1})$ if $x_i+y_i=f_{i-1}(x_1,y_1)$ for all $i$, $2\le i\le l+1$. In this paper we study the diameter of $D(q;\mathbf{f})$ in the special case of monomial digraphs $D(q;m,n)$: $\mathbf{f}=f_1$ and $f_1(x,y)=x^m{y}^n$ for some nonnegative integers $m$ and $n$.