Improving the Delsarte bound

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 $\mu$ 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 $(v,k,\lambda,\mu)$ for strongly regular graphs. Lastly, we provide tables of parameters $(v,k,\lambda,\mu)$ for nonexistent strongly regular graphs with smallest eigenvalue $-4, -5, -6$ or $-7$.