On the spectral gap and the automorphism group of distance-regular graphs

TL;DR

This paper proves that certain distance-regular graphs with a dominant distance are spectral expanders, and uses this to analyze their automorphism groups, confirming conjectures about minimal degrees in specific cases.

Contribution

It establishes a new inequality on intersection numbers, proves spectral expansion for graphs with a dominant distance, and confirms Babai's conjecture for non-geometric primitive distance-regular graphs of bounded diameter.

Findings

Distance-regular graphs with a dominant distance are spectral expanders.

Confirmed Babai's conjecture for non-geometric primitive distance-regular graphs of bounded diameter.

Identified exceptions including cocktail-party graphs when primitivity is not assumed.

Abstract

We prove that a distance-regular graph with a dominant distance is a spectral expander. The key ingredient of the proof is a new inequality on the intersection numbers. We use the spectral gap bound to study the structure of the automorphism group. 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 distance-regular graphs of diameter 2. Babai conjectured that Hamming and Johnson graphs are the only primitive distance-regular graphs of diameter $d\geq 3$ whose automorphism group has sublinear minimal degree. We confirm this conjecture for non-geometric primitive distance-regular graphs of bounded diameter. We also show if the primitivity assumption is removed, then only one additional family of exceptions arises, the cocktail-party graphs. We settle the geometric case in a companion paper.