Monomial graphs and generalized quadrangles

Let Fq be a finite field, where q = p for some odd prime p and integer e ≥ 1. Let f, g ∈ Fq[x, y] be monomials. The monomial graph Gq(f, g) is a bipartite graph with vertex partition P ∪L, P = Fq = L, and (x1, x2, x3) ∈ P is adjacent to [y1, y2, y3] ∈ L if and only if x2 + y2 = f(x1, y1) and x3 + y3 = g(x1, y1). Dmytrenko, Lazebnik, and Williford proved in [6] that if p ≥ 5 and e = 23 for integers a, b ≥ 0, then all monomial graphs Gq(f, g) of girth at least eight are isomorphic to Gq(xy, xy), an induced subgraph of the point-line incidence graph of a classical generalized quadrangle of order q. In this paper, we will prove that for any integer e ≥ 1, there exists a lower bound p0 = p0(e) depending only on the largest prime divisor of e such that the result holds for all p ≥ p0. In particular, we will show that for any integers a, b, c, d, y ≥ 0, the result holds for p ≥ 7 with e = 235; p ≥ 11 with e = 2357; and p ≥ 13 with e = 235711.