Abstract. Let e be a positive integer, p be an odd prime, q = p e , and Fq be the finite field of q elements. Let f, g ∈ Fq[X, Y ]. The graph G = Gq(f, g) is a bipartite graph with vertex partitions P = F 3 q and L = F 3 q , and edges defined as follows:Motivated by some questions in finite geometry and extremal graph theory, Dmytrenko, Lazebnik and Williford conjectured in 2007 that if f and g are both monomials and G has no cycle of length less than eight, then G is isomorphic to the graph Gq(XY, XY 2 ). They proved several instances of the conjecture by reducing it to the property of polynomialsIn this paper we prove the conjecture by obtaining new results on the polynomials A k and B k , which are also of interest on their own.