Asymptotic Delsarte cliques in distance-regular graphs

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."Comment: 10 page

On the automorphism group of a distance-regular graph

The motion of a graph is the minimal degree of its full automorphism group. Babai conjectured that the motion of a primitive distance-regular graph on $n$ vertices of diameter greater than two is at least $n/C$ for some universal constant $C > 0$, unless the graph is a Johnson or Hamming graph. We prove that the motion of a distance-regular graph of diameter $d \geq 3$ on $n$ vertices is at least $Cn/(\log n)^6$ for some universal constant $C > 0$, unless it is a Johnson, a Hamming or a crown graph. This follows using an improvement of an earlier result by Kivva who gave a lower bound on motion of the form $n/c_d$, where $c_d$ depends exponentially on $d$. As a corollary we derive a quasipolynomial upper bound for the automorphism group of a primitive distance-regular graph acting edge-transitively on the graph and on its distance-2 graph. The proofs use elementary combinatorial arguments and do not depend on the classification of finite simple groups.Comment: 16 page