  Let $p$ be an odd prime, $q=p^e$, $e\ge 1$, and $\mathbb{F} = \mathbb{F_q}$
denote the finite field of $q$ elements. Let $f: \mathbb{F}^2\to \mathbb{F}$
and $g: \mathbb{F}^3\to \mathbb{F}$ be functions, and let $P$ and $L$ be two
copies of the 3-dimensional vector space $\mathbb{F}^3$. Consider a bipartite
graph $\Gamma _\mathbb{F} (f, g)$ with vertex partitions $P$ and $L$ and with
edges defined as follows: for every $(p)=(p_1,p_2,p_3)\in P$ and every $[l]=
[l_1,l_2,l_3]\in L$, $\{(p), [l]\} = (p)[l]$ is an edge in $\Gamma _\mathbb{F}
(f, g)$ if $$p_2+l_2 =f(p_1,l_1) \;\;\;\text{and}\;\;\; p_3 + l_3 =
g(p_1,p_2,l_1).$$ Given $\Gamma _\mathbb{F} (f, g)$, is it always possible to
find a function $h:\mathbb{F}^2\to \mathbb{F}$ such that the graph $\Gamma
_\mathbb{F} (f, h)$ with the same vertex set as $\Gamma _\mathbb{F} (f, g)$ and
with edges $(p)[l]$ defined in a similar way by the system $$p_2+l_2
=f(p_1,l_1) \;\;\;\text{and}\;\;\; p_3 + l_3 = h(p_1,l_1),$$ is isomorphic to
$\Gamma _\mathbb{F} (f, g)$ for infinitely many $q$? In this paper we show that
the answer to the question is negative and the graphs
$\Gamma_{\mathbb{F}_p}(p_1\ell_1, p_1\ell_1p_2(p_1 + p_2 + p_1p_2))$ provide
such an example for $p \equiv 1 \pmod{3}$. Our argument is based on proving
that the automorphism group of these graphs has order $p$, which is the
smallest possible order of the automorphism group of graphs of the form
$\Gamma_{\mathbb{F}}(f, g)$.
