Monomial graphs and generalized quadrangles

Let Fq be a finite field, where q=pe 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, , 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 (2007) proved in [5] that if p⩾5 and e=2a3b for integers a,b⩾0, then all monomial graphs Gq(f,g) of girth at least eight are isomorphic to Gq(xy,xy2), 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=2a3b5c; p⩾11 with e=2a3b5c7d; and p⩾13 with e=2a3b5c7d11y.