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

Abstract

One of the central results in the representation theory of distance-regular graphs classifies distance-regular graphs with µ ≥ 2 and second largest eigenvalue θ 1 = b 1 −1.In this paper we give a classification under the (weaker) approximate eigenvalue constraint θ 1 ≥ (1 − ε)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.