Asymptotic Delsarte cliques in distance-regular graphs

Abstract

We give a new bound on the parameter $\lambda$ (number of common neighbors of a pair of adjacent vertices) in a distance-regular graph G, improving and generalizing bounds for strongly regular graphs by Spielman (1996) and Pyber (2014. arXiv:1409.3041). The new bound is one of the ingredients of recent progress on the complexity of testing isomorphism of strongly regular graphs (Babai et al. 2013). The proof is based on a clique geometry found by Metsch (Des Codes Cryptogr 1(2):99–116, 1991) under certain constraints on the parameters. We also give a simplified proof of the following asymptotic consequence of Metsch’s result: If $k\mu = o(\lambda ^2)$, then each edge of G belongs to a unique maximal clique of size asymptotically equal to $\lambda$, and all other cliques have size $o(\lambda )$. Here k denotes the degree and $\mu$ the number of common neighbors of a pair of vertices at distance 2. We point out that Metsch’s cliques are “asymptotically Delsarte” when $k\mu = o(\lambda ^2)$, so families of distance-regular graphs with parameters satisfying $k\mu = o(\lambda ^2)$ are “asymptotically Delsarte-geometric.”