On algebraically defined graphs in three dimensions

Let F be a field of characteristic zero or of a positive odd characteristic p. For a polynomial f ∈ F[x, y], we define a graph ΓF(xy, f) to be a bipartite graph with vertex partition P ∪L, P = F = L, and (p1, p2, p3) ∈ P is adjacent to [l1, l2, l3] ∈ L if and only if p2 + l2 = p1l1 and p3 + l3 = f(p1, l1). If f = xy, the graph ΓF(xy, xy ) has the length of a shortest cycle (the girth) equal to eight, and it is isomorphic to an induced subgraph of a classical generalized quadrangle. In this paper, we will prove that when F is the field of complex numbers, then every girth eight graph ΓF(xy, f) is isomorphic to ΓF(xy, xy ). Furthermore, we will prove a similar result over infinitely many finite fields F.