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.”