Augmenting the Delsarte bound: A forbidden interval for the order of maximal cliques in strongly regular graphs. (October 2021)

• Record Type: Journal Article
• Authors: Greaves, Gary R.W.; Koolen, Jack H.; Park, Jongyook
• Is Part Of: European journal of combinatorics. Volume 97(2021)
• Publication Date: 2021-10
• Subjects: Combinatorial analysis -- Periodicals; Analyse combinatoire -- Périodiques; Combinatorial analysis; Periodicals; Electronic journals; 511.6
• DOI: 10.1016/j.ejc.2021.103384
• Languages: English
• ISSNs: 0195-6698

Abstract

Abstract: In this paper, we study the order of a maximal clique in an amply regular graph with a fixed smallest eigenvalue by considering a vertex that is adjacent to some (but not all) vertices of the maximal clique. As a consequence, we show that if a strongly regular graph contains a Delsarte clique, then the parameter μ is either small or large. Furthermore, we obtain a cubic polynomial that assures that a maximal clique in an amply regular graph is either small or large (under certain assumptions). Combining this cubic polynomial with the claw-bound, we rule out an infinite family of feasible parameters ( v, k, λ, μ ) for strongly regular graphs. Lastly, we provide tables of parameters ( v, k, λ, μ ) for nonexistent strongly regular graphs with smallest eigenvalue − 4, − 5, − 6 or − 7 .