On the uniqueness of some girth eight algebraically defined graphs

Let FF be a field. For a polynomial f∈F[x,y]f∈F[x,y], we define a bipartite graph ΓF(f)ΓF(f) with vertex partition P∪LP∪L, P=F3=LP=F3=L, and (p1,p2,p3)∈P(p1,p2,p3)∈P is adjacent to [l1,l2,l3]∈L[l1,l2,l3]∈L if and only if p2+l2=p1l1andp3+l3=f(p1,l1). It is known that the graph ΓF(xy2)ΓF(xy2) has no cycles of length less than eight. The main result of this paper is that ΓF(xy2)ΓF(xy2) is the only graph ΓF(f)ΓF(f) with this property when FF is an algebraically closed field of characteristic zero; i.e. over such a field FF, every graph ΓF(f)ΓF(f) with no cycles of length less than eight is isomorphic to ΓF(xy2)ΓF(xy2). We also prove related uniqueness results for some polynomials ff over infinite families of finite fields.