On the automorphism groups of strongly regular graphs I

Abstract

We derive 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(n1/5)) (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((ln n)2ln ln n). (The exceptions admit trivial linear-time GI testing.) The design of isomorphism tests for various classes of structures is intimately connected with the study of the automorphism groups of those structures. We include a brief survey of these connections, starting with an 1869 paper by Jordan on trees. In particular, our result 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 remains 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. We state a purely mathematical conjecture that could bring us closer to finding the right kind of hierarchy. We also outline the broader GI context, and state conjectures in terms of "primitive coherent configurations." These are generalizations of s.r. graphs, relevant to the general GI problem. Another consequence of the main result is the strongest argument to date against GI-completeness of s.r. graphs: we prove that no polynomial-time categorical reduction of GI to isomorphism of s.r. graphs is possible. All known reductions between isomorphism problems of various classes of structures fit into our notion of "categorical reduction." The proof of the main result is elementary; it is based on known results in spectral graph theory and on a 1987 lemma on permutations by Ákos Seress and the author.