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 θ1=b1−1. In this paper we give a classification under the (weaker) approximate eigenvalue constraint θ1≥(1−ε)b1 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 of bounded diameter. This conjecture asserts that if X is a primitive distance-regular graph with n vertices, and X is not a Hamming graph or a Johnson graph, then the automorphism group Aut(X) has minimal degree ≥cn for some constant c>0. It follows that Aut(X) satisfies strong structural constraints.