A new family of algebraically defined graphs with small automorphism group (Q2121782)

Summary: Let \(p\) be an odd prime, \(q=p^e, e \geqslant 1\), 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 \[p_2+l_2 =f(p_1,l_1) \;\;\;\text{and}\;\;\; p_3 + l_3 = g(p_1,p_2,l_1).\] The following question appeared in [\textit{B. Nassau}, Studies of some classes of algebraically defined graphs. Newark, DE: University of Delaware (PhD Thesis) (2020)]: 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 \(\Gamma_{\mathbb{F}}(f, g)\).