Let e be a positive integer, p be an odd prime, q = p e , and F q be the finite field of q elements. Let f 2 , f 3 ∈ F q [x, y]. The graph G = G q (f 2 , f 3) is a bipartite graph with vertex partitions P = F 3 q and L = F 3 q , and edges defined as follows: a vertex (p) = (p 1 , p 2 , p 3) ∈ P is adjacent to a vertex [l] = [l 1 , l 2 , l 3 ] if and only if p 2 + l 2 = f 2 (p 1 , l 1) and p 3 + l 3 = f 3 (p 1 , l 1). 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 f 2 and f 3 are monomials, we call G a monomial graph. We show that for p 5, and e = 2 a 3 b , a monomial graph of girth at least eight has to be isomorphic to the graph G q (xy, xy 2), 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 G q (xy, x k y 2k), with 1 k (q − 1)/2 and satisfying several other strong conditions. These conditions imply that k = 1 for all q 10 10. In particular, for a given positive integer k, the graph G q (xy, x k y 2k) can be of girth eight only for finitely many odd characteristics p.

Let q be a prime power, F q be the field of q elements, and k; m be positive integers. A bipartite graph G ¼ G q ðk; mÞ is defined as follows. The vertex set of G is a union of two copies P and L of two-dimensional vector spaces over F q , with two vertices ðp 1 ; p 2 Þ 2 P and ½ l 1 ; l 2 2 L being adjacent if and only if p 2 þ l 2 ¼ p

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