Structure, Automorphisms, and Isomorphisms of Regular Combinatorial Objects

Abstract

We develop new structure theory for highly regular combinatorial objects, including Steiner designs, strongly regular graphs, and coherent configurations. As applications, we make progress on old problems in algebraic combinatorics and the theory of permutation groups, and break decades-old barriers on the complexity of the algorithmic Graph Isomorphism problem. A central aspect of our structural contributions is the discovery of clique geometries in regular structures. A second aspect is bounds on the rate of expansion of small sets. In the case of Steiner designs, we give a $n^{O(\log n)}$ bound on the number of automorphisms where $n$ is the number of points. This result is nearly optimal in two ways: it essentially matches the number of automorphisms in affine or projective space, and we show that the bound does not extend to the broader class of balanced incomplete block designs. The line-graphs of Steiner designs are strongly regular graphs, and in fact are one of the cases of Neumaier's classification of strongly regular graphs. We bound the number of reconstructions of a Steiner design from its line-graph in order to apply our automorphism bound for Steiner designs to this class of strongly regular graphs, and show that this class of strongly regular graphs has at most $\exp(\tilde{O}(v^{1/14}))$ automorphisms, where $v$ is the number of vertices and the $\tilde{O}$ hides polylogarithmic factors. We give an $\exp(\tilde{O}(1 + \lambda/\mu))$ bound on the number of automorphisms of any nontrivial $\SR(v,\rho,\lambda,\mu)$ strongly regular graph. (Here, $v$ is the number of vertices, $\rho$ is the valency, and $\lambda$ and $\mu$ are the number of common neighbors of a pair of adjacent and nonadjacent vertices, respectively.) As a consequence, we obtain a quasipolynomial bound on the number of automorphisms when $\rho = \Omega(v^{5/6})$. In further study of the structure of the automorphism groups of $\SR(v,\rho,\lambda,\mu)$ graphs, we find a $\Gamma_\mu$ subgroup of index $v^{O(\log v)}$ (i.e., a subgroup of index $v^{O(\log v)}$ for which all composition factors are subgroups of $S_{\mu}$) with known exceptions. In combination with our bound on the number of automorphisms and an earlier bound due to Spielman, we find a $\Gamma_d$ subgroup of the automorphism group of index $v^d$, where $d = \tilde{O}(v^{1/5})$, again with known exceptions. We classify the primitive coherent configurations with not less than $\exp(\tilde{O}(v^{1/3}))$ automorphisms, where $v$ is the number of vertices. As a corollary to our combinatorial classification result, we infer a classification of large primitive permutation groups, previously known only through the Classification of Finite Simple Groups. As a consequence of the combinatorial structure underlying our bounds for the automorphism groups, we give corresponding bounds for the time-complexity of deciding isomorphism. When we bound the order of the automorphism group, our time-complexity bounds are identical to the bounds on the order. From our study of the composition factors of automorphism groups of strongly regular graphs, we obtain a $v^{\mu + O(\log v)}$ and a $\exp(\tilde{O}(v^{1/5}))$ bound on the time-complexity of deciding isomorphism of strongly regular graphs.