On the Automorphism Groups of Strongly Regular Graphs

We derive strong structural constraints on the automorphism groups of strongly regular (s. r.) graphs, giving a surprisingly strong answer to a decades-old problem, with tantalizing implications to testing isomorphism of s. r. graphs, and raising new combinatorial challenges. S. r. graphs, while not believed to be Graph Isomorphism (GI) complete, have long been recognized as hard cases for GI, and, in this author’s view, present some of the core difficulties of the general GI problem. Progress on the complexity of testing their isomorphism has been intermittent (Babai 1980, Spielman 1996, BW & CST (STOC’13) and BCSTW (FOCS’13)), and the current best bound is exp(Õ(n)) (n is the number of vertices). Our main result is that if X is a s. r. graph then, with straightforward exceptions, the degree of the largest alternating group involved in the automorphism group Aut(X) (as a quotient of a subgroup) is O((lnn)/ ln lnn). (The exceptions admit trivial linear-time GI testing.) This result greatly amplifies the potential of Luks’s divide-and-conquer methods (1980) to be applicable to testing isomorphism of s. r. graphs in quasipolynomial time. The challenge is to find a hierarchy of combinatorial substructures through which this potential can be realized. We expect that the generality of our result will help in this regard; the result applies not only to s. r. graphs but to all graphs with strong spectral expansion and with a relatively small number of common neighbors for every pair of vertices.