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

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 ≥ cn, 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 ≥ 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.

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

“…Finally, we combine the results of this paper and of [23] to get the proof of Babai's conjecture (Conjecture 1.10). From [23], in addition to Theorems 1.12 and 5.2, we need the following observation. Proof.…”

“…(The imprimitive case admits one more class of exceptions, the cocktail-party graphs). A large portion of the proof relies on the results we prove in [23].…”

“…In [23] we reduced the case of imprimitive distance-regular graphs to confirming the conjecture for primitive graphs. Moreover, in one of the main results of [23] we show that it is sufficient to study geometric distance-regular graphs with bounded smallest eigenvalue, which is the case we settle in this paper. Remark 1.13.…”

“…The reason we split the proof of Theorem 1.11 between two papers is the different nature of techniques and results obtained. In particular, in [23] we prove a spectral gap bound for distance-regular graphs which have a dominant distance. We use it as one of the ingredients in the analysis of µ = 1 case in this paper (see Theorem 5.2).…”

“…For this, it is sufficient to show that X has a linear in k spectral gap. First, we bound the zero-weight spectral radius of a geometric distance-regular graph using the following lemma proven in [23]. Using the relationship between the spectrum of the geometric graph X and its dual graph X we get the following corollary.…”