  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 $-7$.
