An Isomorphism Criterion for Monomial Graphs

Let q be a prime power, Fq be the field of q elements, and k, m be positive integers. A bipartite graph G = Gq(g, h, k,m) is defined as follows. The vertex set of G is a union of two copies P and L of two-dimensional vector spaces over Fq, with two vertices (p1, p2) ∈ P and [ l1, l2 ] ∈ L being adjacent if and only if p2 + l2 = p1l 1 . We prove that graphs Gq(k, m) and Gq′(k,m) are isomorphic if and only if q = q′ and {gcd(k, q − 1), gcd(m, q − 1)} = {gcd(k′, q − 1), gcd(m′, q − 1)} as multisets. The proof is based on counting the number of complete bipartite subgraphs in the graphs.