Nonisomorphic two-dimensional algebraically defined graphs over R

For f: R-2 -> R, let Gamma(R)(f) be a two-dimensional algebraically defined graph, that is, a bipartite graph where each partite set is a copy of R-2 and two vertices (a, a(2)) and [x, x(2)] are adjacent if and only if a(2) + x(2) = f(a, x). It is known that Gamma(R)(XY) has girth 6 and can be extended to the point-line incidence graph of the classical real projective plane. However, it was unknown whether there exists f is an element of R[X, Y] such that Gamma(R)(f) has girth 6 and is nonisomorphic to Gamma(R)(XY). This paper answers this question affirmatively and thus provides a construction of a nonclassical real projective plane. This paper also studies the diameter and girth of Gamma(R)(f) for families of bivariate functions f.