A characterization of Johnson and Hamming graphs and proof of Babai's conjecture

TL;DR

This paper classifies geometric distance-regular graphs with approximate eigenvalue constraints and confirms Babai's conjecture on the minimal degree of automorphism groups of such graphs.

Contribution

It extends the classification of distance-regular graphs to approximate eigenvalue conditions and proves Babai's conjecture for these graphs.

Findings

Classification of geometric distance-regular graphs with approximate eigenvalue bounds

Confirmation of Babai's conjecture on automorphism group degrees

New insights into the spectral properties of distance-regular graphs

Abstract

One of the central results in the representation theory of distance-regular graphs classifies distance-regular graphs with $\mu\geq 2$ and second largest eigenvalue $\theta_1= b_1-1$. In this paper we give a classification under the (weaker) approximate eigenvalue constraint $\theta_1\geq (1-\varepsilon)b_1$ for the class of geometric distance-regular graphs. As an application, we confirm Babai's conjecture on the minimal degree of the automorphism group of distance-regular graphs.