On the Automorphism Groups of Strongly Regular Graphs I László Babai University of Chicago 1100 E 58th St Chicago, IL 60637 [email protected] To the memory of Akos´ Seress (1958{2013) in terms of \primitive coherent configurations." These are generalizations of s. r. graphs, relevant to the general GI problem. ABSTRACT Another consequence of the main result is the strongest We derive structural constraints on the automorphism groups argument to date against GI-completeness of s. r. graphs: of strongly regular (s. r.) graphs, giving a surprisingly strong we prove that no polynomial-time categorical reduction of answer to a decades-old problem, with tantalizing implica- GI to isomorphism of s. r. graphs is possible. All known tions to testing isomorphism of s. r. graphs, and raising new reductions between isomorphism problems of various classes combinatorial challenges. of structures fit into our notion of \categorical reduction." S. r. graphs, while not believed to be Graph Isomorphism The proof of the main result is elementary; it is based on (GI) complete, have long been recognized as hard cases for known results in spectral graph theory and on a 1987 lemma GI, and, in this author's view, present some of the core diffi- on permutations by Akos´ Seress and the author. culties of the general GI problem. Progress on the complex- ity of testing their isomorphism has been intermittent (Babai Categories and Subject Descriptors 1980, Spielman 1996, BW & CST (STOC'13) and BCSTW Mathematics of computing [Discrete mathematics]: Graph (FOCS'13)), and the current best bound is exp(Oe(n1=5)) theory; Theory of computation [Design and analysis of (n is the number of vertices). algorithms]: Graph algorithms analysis Our main result is that if X is a s. r. graph then, with straightforward exceptions, the degree of the largest alter- nating group involved in the automorphism group Aut(X) Keywords 2 (as a quotient of a subgroup) is O((ln n) = ln ln n). (The graphs, groups, automorphism groups, algorithms, isomor- exceptions admit trivial linear-time GI testing.) phism testing, strongly regular graphs The design of isomorphism tests for various classes of structures is intimately connected with the study of the au- tomorphism groups of those structures. We include a brief 1. INTRODUCTION survey of these connections, starting with an 1869 paper by In just a few lines and from mostly known igredients we Jordan on trees. derive a surprisingly strong and unanticipated answer to In particular, our result amplifies the potential of Luks's a decades-old mathematical question with implications to divide-and-conquer methods (1980) to be applicable to test- graph isomorphism testing, the combinatorics of highly reg- ing isomorphism of s. r. graphs in quasipolynomial time. ular configurations, and the theory of permutation groups. The challenge remains to find a hierarchy of combinatorial Accordingly, this paper is short on proofs and long on mo- substructures through which this potential can be realized. tivation. The proofs, albeit simple, draw on diverse sources: We expect that the generality of our result will help in this two distinct areas of spectral graph theory (expansion, where regard; the result applies not only to s. r. graphs but to all the focus is on inequalities; and the spectral aspects of strong graphs with strong spectral expansion and with a relatively regularity, where identities play a central role), a lemma on small number of common neighbors for every pair of ver- permutations that initially arose in the context of parallel tices. We state a purely mathematical conjecture that could algorithms and the diameter of permutation groups; and a bring us closer to finding the right kind of hierarchy. We combinatorial lemma about strongly regular graphs, initially also outline the broader GI context, and state conjectures devised in the context of isomorphism testing. We start with describing the main results, and discuss the Permission to make digital or hard copies of all or part of this work for motivation in a subsequent section. In a brief \Outlook" sec- personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear tion (Sec. 8) we outline the broader graph isomorphism (GI) this notice and the full citation on the first page. Copyrights for components context and state relevant conjectures in terms of \primitive of this work owned by others than ACM must be honored. Abstracting with coherent configurations." credit is permitted. To copy otherwise, or republish, to post on servers or to Nothing but the most basic group theory is required for redistribute to lists, requires prior specific permission and/or a fee. Request the main results (the notion of the alternating group and of permissions from [email protected]. subgroups and quotient groups). Elements of the theory of ITCS’14, January 12–14, 2014, Princeton, NJ, USA Copyright 2014 ACM 978-1-4503-2698-8/14/01 permutation groups are required for the algorithmic motiva- http://dx.doi.org/10.1145/2554797.2554830 ...$15.00. tion, specifically the notion of primitive permutation groups. These concepts have been fundamental to the area of GI The best previously known bound is θ(Aut(X)) = Oe(n1=5), testing ever since Gene Luks's seminal 1980 paper [46]. We inferable from [9] (see Section B in the Appendix); prior to review basic permutation group concepts in the Appendix 2013, the best known bound was Oe(n1=3), inferable from (Sec. A). The reader may understand much of the motiva- Spielman [59]. (The tilde hides polylogarithmic factors.) tion by having just the vague notion that \primitive permu- One more piece of terminology will come in handy. tation groups" are those permutation groups where natural \divide and conquer" breaks down. Definition 3. Let X be a regular graph of degree k. Let k = ξ1 ≥ ξ2 ≥ · · · ≥ ξn denote the eigenvalues of the adja- 1.1 The main results cency matrix of X. Set ξ = ξ(X) = maxfjξij j 2 ≤ i ≤ ng. A graph X is strongly regular with parameters (n; k; λ, µ) We call this quantity the zero-weight spectral radius of X. (It is the spectral radius of the adjacency operator restricted if X has n vertices, every vertex has degree k, each pair of P adjacent vertices has λ common neighbors, and each pair of to the subspace xi = 0:) non-adjacent vertices has µ common neighbors. We shall actually prove the following more general result. A group G is said to involve the group H if H =∼ L=N for some N/L ≤ G (quotient of a subgroup). Theorem 4. Let X be a regular graph of degree k with Next we introduce a term motivated by the asymptotic zero-weight spectral radius of ξ. Suppose every pair of ver- theory of primitive permutation groups (see Theorem 6) that tices in X has at most q common neighbors. Assume q +ξ < will be convenient to use in the statement of our main re- k. Then the thickness of Aut(X) is at most sults. (ln n)2 1 + o(1) · : (1) 2 ln ln n q+ξ 2 Definition 1. The thickness θ(G) of a group G is the great- 1 − k est t such that the alternating group At is involved in G. (Here the o(1) term goes to zero as n ! 1 uniformly re- We note that this is not standard terminology, but Peter gardless of the other parameters.) Cameron, one of the architects of asymptotic group theory, This will be proved in Section 4. agreed to use this term in the future1. To infer Theorem 2 from Theorem 4, we shall need to We shall say that the strongly regular graph X is trivial show that for non-trivial, non-graphic s. r. graphs, q + ξ is if X or its complement is disconnected. In this case, X or bounded away from k. Since the complement of a s. r. graph its complement is the disjoint union of cliques of equal size. is s. r., it suffices to prove this under the assumption that our We shall say that X is graphic if X or its complement is the s. r. graphs have degree k ≤ (n−1)=2. In fact, we shall need line graph of a graph. (The vertices of the line graph L(Y ) to assume k < n=4. For the cases k ≥ n=4 we shall take a of the graph Y correspond to the edges of Y ; two vertices more direct approach. of L(Y ) are adjacent in L(Y ) if the corresponding edges of Y share a vertex.) The line graph L(Y ) is s. r. exactly if Theorem 5. Let X be a non-trivial, non-graphic strongly v regular graph of degree k with n ≥ 29 vertices and zero- Y is either a complete graph Kv (n = ) or a complete 2 weight spectral radius of ξ. Suppose every pair of vertices bipartite graph K (with equal parts; n = v2). v;v in X has at most q common neighbors. Assume k ≤ n=4. We note that it is straightforward to recognize trivial and Then q + ξ < 7k=8. graphic s. r. graphs and to test their isomorphism in linear time. We also note that the automorphism groups ofp these This will be proved in Section 5.1. families of s. r. graphs have large thickness: at least n in Theorems 2 and 4 will immediately follow from bounds each case. we establish on the order of automorphisms (Prop. 13 and Theorem 20, respectively). Specifically, we prove that the Theorem 2.