On the Girth of Three-Dimensional Algebraically Defined Graphs with Multiplicatively Separable Functions

For a field F and functions f, g, h, j : F -> F, we define Gamma(F)(f(X)h(Y), g(X)j(Y)) to be a bipartite graph where each partite set is a copy of F-3, and a vertex (a, a(2), a(3)) in the first partite set is adjacent to a vertex [x, x(2,) x(3)] in the second partite set if and only if a(2) + x(2) = f(a)h(x) and a(3) + x(3) = g(a)j(x). In this paper, we completely classify all such graphs by girth in the case h = j (subject to some mild restrictions on h). We also present a partial classification when h not equal j and provide some applications.