On the automorphism groups of distance-regular graphs and rank-4 primitive coherent configurations

TL;DR

This paper extends Babai's lower bound on the minimal degree of automorphism groups from strongly regular graphs to primitive coherent configurations of rank 4 and certain distance-regular graphs, using structural and spectral methods.

Contribution

It generalizes a key lower bound on automorphism group minimal degree from rank 3 to rank 4 configurations and some distance-regular graphs, broadening structural understanding.

Findings

Automorphism groups of rank-4 primitive coherent configurations have minimal degree at least a constant fraction of the number of vertices.

The result applies to non-geometric distance-regular graphs with bounded diameter.

Structural and spectral techniques are used to establish these bounds.

Abstract

The minimal degree of a permutation group $G$ is the minimum number of points not fixed by non-identity elements of $G$. Lower bounds on the minimal degree have strong structural consequences on $G$. In 2014 Babai proved that the automorphism group of a strongly regular graph with $n$ vertices has minimal degree $\geq c n$, with known exceptions. Strongly regular graphs correspond to primitive coherent configurations of rank 3. We extend Babai's result to primitive coherent configurations of rank 4. We also show that the result extends to non-geometric distance-regular graphs of bounded diameter. The proofs combine structural and spectral methods.