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

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. Ministry of Education (MOE) Gary Greaves is partially supported by the Singapore Ministry of Education Academic Research Fund (Tier 1); grant numbers: RG29/18 and RG21/20. Jack H. Koolen is partially supported by the National Natural Science Foundation of China (No. 12071454) and Anhui Initiative in Quantum Information Technologies (No. AHY150000). And the research was partially supported by the project ‘‘Analysis and Geometry on Bundles’’ of Ministry of Science and Technology of the People’s Republic of China. Jongyook Park is partially supported by Basic Science Research Program through the National Research Foundation of Korea funded by Ministry of Education (NRF-2017R1D1A1B03032016) and the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (NRF-2020R1A2C1A01101838).