Nonisomorphic two‐dimensional algebraically defined graphs over R

• Published in: Journal of graph theory Vol. 108; no. 1; pp. 50 - 64
• Main Authors: Kronenthal, Brian G, Miller, Joe, Nash, Alex, Roeder, Jacob, Samamah, Hani, Wong, Tony W H
• Format: Journal Article
• Language: English
• Published: Hoboken Wiley Subscription Services, Inc 01.01.2025
• Subjects: Apexes; Bivariate analysis; Graph theory
• ISSN: 0364-9024; 1097-0118
• DOI: 10.1002/jgt.23161

For f:R2→R, let ΓR(f) be a two‐dimensional algebraically defined graph, that is, a bipartite graph where each partite set is a copy of R2 and two vertices (a,a2) and [x,x2] are adjacent if and only if a2+x2=f(a,x). It is known that ΓR(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∈R[X,Y] such that ΓR(f) has girth 6 and is nonisomorphic to ΓR(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 ΓR(f) for families of bivariate functions f.