Asymptotic Delsarte cliques in distance-regular graphs

TL;DR

This paper introduces a new bound on the parameter λ in distance-regular graphs, improving previous bounds and revealing asymptotic Delsarte properties, with implications for graph isomorphism testing.

Contribution

It provides a new bound on λ, simplifies proofs of asymptotic clique structures, and links distance-regular graphs to Delsarte geometry under certain parameter conditions.

Findings

New bound on λ improves previous results.

Edges belong to unique maximal cliques asymptotically.

Distance-regular graphs with certain parameters are asymptotically Delsarte-geometric.

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). The new bound is one of the ingredients of recent progress on the complexity of testing isomorphism of strongly regular graphs (Babai, Chen, Sun, Teng, Wilmes 2013). The proof is based on a clique geometry found by Metsch (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."