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arXiv · 2012.09391

Improving the Delsarte bound

Gary R. W. Greaves·Jack H. Koolen·Jongyook Park
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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 $-7$.

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Gary R. W. Greaves, Jack H. Koolen, Jongyook Park. 2020-12-17. Improving the Delsarte bound. https://arxiv.org/abs/2012.09391

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