On monomial graphs of girth eight

Let e be a positive integer, p be a odd prime, q = p e, and Fq be the finite field of q elements. Let f2, f3 ∈ Fq[x, y]. The graph G = Gq(f2, f3) is a bipartite graph with vertex partitions P = Fq and L = Fq , and edges defined as follows: a vertex (p) = (p1, p2, p3) ∈ P is adjacent to a vertex [l] = [l1, l2, l3] if and only if p2 + l2 = f2(p1, l1) and p3 + l3 = f3(p1, l1). Motivated by some questions in finite geometry and extremal graph theory, we ask when G has no cycle of length less than eight, i.e., has girth at least eight. When f2 and f3 are monomials, we call G a monomial graph. We show that for p ≥ 5, and e = 2a3b, a monomial graph of girth at least eight has to be isomorphic to graph Gq(xy, xy2), which is an induced subgraph of the classical generalized quadrangle W (q). For all other e, we show that a monomial graph is isomorphic to a graph Gq(xy, xky2k), with 1 ≤ k ≤ (q − 1)/2 and satisfying several other strong conditions. These conditions imply that k = 1 for all q ≤ 1010. In particular, for a given positive integer k, graph Gq(xy, xky2k) can be of girth eight only for finitely many odd characteristics p.