On the uniqueness of some girth eight algebraically defined graphs, Part II

Let IF be a field. For a polynomial f is an element of F[x, y] and positive integers k and m, we define a bipartite graph Gamma(F)(x(k)y(m), f) with vertex partition P U L, where P and L are two copies of F3, and (p(1), P-2, P-3) E P is adjacent to [l(1), l(2), l(3)] is an element of L if and only if P-2 + l(2) = P(1)(k)l(1)(m) and P-3 + l(3) =f(P-1, l(1)) It is known that T'F(xy, xy(2)) has no cycles of length less than eight. The main result of this paper is that FF(xY, xy(2)) is the only graph FF(xkym,f) with this property when IF is an algebraically closed field of characteristic zero; i.e. over such a field F, every graph Gamma(F)(x(k)y(m),f) with no cycles of length less than eight is isomorphic to TF(xy, xy2). We also prove related uniqueness results over infinite families of finite fields. (C) 2018 Elsevier B.V. All rights reserved.