Nonisomorphic two‐dimensional algebraically defined graphs over R R - Kronenthal, Brian G.; Miller, Joe; Nash, Alex; Roeder, Jacob; Samamah, Hani; Wong, Tony W. H. (Journal of graph theory, 2025)
For f : R 2 → R, let Γ 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 Γ R ( X Y ) 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 ∈ R [ X , Y ] such that Γ R ( f ) has girth 6 and is nonisomorphic to Γ R ( X Y ). 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 Γ R ( f ) for families of bivariate functions f.