Augmenting the Delsarte bound: a forbidden interval for the order of maximal cliques in strongly regular graphs

• Main Authors: Greaves, Gary Royden Watson, Koolen, Jack H., Park, Jongyook
• Other Authors: School of Physical and Mathematical Sciences
• Format: Article
• Language: English
• Published: 2022
• Subjects: Science::Mathematics; Delsarte Clique; Eigenvalue
• Online Access: https://hdl.handle.net/10356/159972
• Institution: Nanyang Technological University

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.