Journal of Algebraic Combinatorics

, Volume 43, Issue 4, pp 771–782 | Cite as

Asymptotic Delsarte cliques in distance-regular graphs

  • László BabaiEmail author
  • John Wilmes


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.”


Distance-regular graphs Clique Clique geometry Delsarte clique Asymptotic analysis 



The authors wish to acknowledge the inspiration from their joint work with Xi Chen, Xiaorui Sun, and Shang-Hua Teng on the isomorphism problem for strongly regular graphs.


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© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Departments of Computer Science and MathematicsUniversity of ChicagoChicagoUSA

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