Isomorphism criterion for monomial graphs

• Published in: Journal of graph theory Vol. 48; no. 4; pp. 322 - 328
• Main Authors: Dmytrenko, Vasyl, Lazebnik, Felix, Viglione, Raymond
• Format: Journal Article
• Language: English
• Published: Hoboken Wiley Subscription Services, Inc., A Wiley Company 01.04.2005
• Subjects: algebraic constructions; graph isomorphism; number of complete bipartite subgraphs
• ISSN: 0364-9024; 1097-0118
• DOI: 10.1002/jgt.20055

Let q be a prime power, 𝔽q be the field of q elements, and k, m be positive integers. A bipartite graph G = Gq(k, m) is defined as follows. The vertex set of G is a union of two copies P and L of two‐dimensional vector spaces over 𝔽q, with two vertices (p1, p2) ∈ P and [ l1, l2] ∈ L being adjacent if and only if p2 + l2 = p 1kl 1m. We prove that graphs Gq(k, m) and Gq′(k′, m′) are isomorphic if and only if q = q′ and {gcd (k, q − 1), gcd (m, q − 1)} = {gcd (k′, q − 1),gcd (m′, q − 1)} as multisets. The proof is based on counting the number of complete bipartite INFgraphs in the graphs. © 2005 Wiley Periodicals, Inc. J Graph Theory 48: 322–328, 2005