Nonisomorphic two‐dimensional algebraically defined graphs over R <math xmlns="http://www.w3.org/1998/Math/MathML" altimg="urn:x-wiley:03649024:media:jgt23161:jgt23161-math-0001" wiley:location="equation/jgt23161-math-0001.png"> 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: 01.01.2025
• Subjects: algebraically defined graph; girth; nonisomorphic; real projective plane
• ISSN: 0364-9024; 1097-0118
• DOI: 10.1002/jgt.23161

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.