A New Family of Algebraically Defined Graphs with Small Automorphism Group

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