On the automorphism groups of strongly regular graphs I

Babai, László

doi:10.1145/2554797.2554830

Cited by 17 publications

(52 citation statements)

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“…We follow Babai's approach [3]. In that paper, Babai used a combination of an old combinatorial tool [1] and an introduced spectral tool.…”

Section: Motion Lower Bounds: the Main Toolsmentioning

confidence: 99%

“…Lemma 1.14 (Babai,[3,Proposition 12]). Let X be a regular graph of degree k on n vertices with the zero-weight spectral radius ξ.…”

Section: Spectral Toolmentioning

confidence: 99%

“…The thickness θ(G) of a group G is the greatest t for which the alternating group A t is involved as a quotient group of a subgroup of G (the term was coined in [3]). Lower bounds on the minimal degree have strong structural consequences on G. In 1934 Wielandt [33] showed that a linear lower bound on the minimal degree implies a logarithmic upper bound on the thickness of the group.…”

Section: Introductionmentioning

confidence: 99%

“…Switching from symmetry assumptions to regularity, in 2014 Babai [3,4] proved that the minimal degree of the automorphism group of a strongly regular graph is linear in the number of vertices, with known exceptions. Definition 1.1.…”

Section: Introductionmentioning

confidence: 99%

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On the automorphism groups of rank-4 primitive coherent configurations

Kivva¹

2021

Preprint

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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. Babai conjectured that if a primitive coherent configuration with n vertices is not a Cameron scheme, then its automorphism group has minimal degree ≥ cn for some constant c > 0. In 2014, Babai proved the desired lower bound on the minimal degree of the automorphism groups of strongly regular graphs, thus confirming the conjecture for primitive coherent configurations of rank 3.In this paper, we extend Babai's result to primitive coherent configurations of rank 4, confirming the conjecture in this special case. The proofs combine structural and spectral methods.

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“…We follow Babai's approach [3]. In that paper, Babai used a combination of an old combinatorial tool [1] and an introduced spectral tool.…”

Section: Motion Lower Bounds: the Main Toolsmentioning

confidence: 99%

“…Lemma 1.14 (Babai,[3,Proposition 12]). Let X be a regular graph of degree k on n vertices with the zero-weight spectral radius ξ.…”

Section: Spectral Toolmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

On the automorphism groups of rank-4 primitive coherent configurations

Kivva¹

2021

Preprint

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“…Progress on Babai's conjecture is significant in part because the detailed classification of large primitive permutation groups due to Cameron [12] implies, as pointed out in [4], that isomorphism of CCs with primitive automorphism groups of order greater than exp(n ε ) can be tested in polynomial time. Hence, confirmation of the conjecture would raise hope that an analysis of I/R could give a subexponential-time isomorphism test for PCCs.…”

Section: Individualization/refinement and The Automorphism Boundmentioning

confidence: 99%

Faster Canonical Forms for Primitive Coherent Configurations

Sun

Wilmes

2015

Proceedings of the Forty-Seventh Annual ACM Symposium on Theory of Computing

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Coherent configurations (CCs) are highly regular colorings of the set of ordered pairs of a "vertex set"; each color represents a "constituent digraph." CCs arise in the study of permutation groups, combinatorial structures such as partially balanced designs, and the analysis of algorithms; their history goes back to Schur in the 1930s. A CC is primitive (PCC) if all its constituent digraphs are connected.We address the problem of classifying PCCs with large automorphism groups. This project was started in Babai's 1981 paper in which he showed that only the trivial PCC admits more than exp( O(n 1/2 )) automorphisms. (Here, n is the number of vertices and the O hides polylogarithmic factors.)In the present paper we classify all PCCs with more than exp( O(n 1/3 )) automorphisms, making the first progress on Babai's conjectured classification of all PCCs with more than exp(n ǫ ) automorphisms.A corollary to Babai's 1981 result solved a then 100-year-old problem on primitive but not doubly transitive permutation groups, giving an exp( O(n 1/2 )) bound on their order. In a similar vein, our result implies an exp( O(n 1/3 )) upper bound on the order of such groups, with known exceptions. This improvement of Babai's result was previously known only through the Classification of Finite Simple Groups (Cameron, 1981), while our proof, like Babai's, is elementary and almost purely combinatorial.Our analysis relies on a new combinatorial structure theory we develop for PCCs. In particular, we demonstrate the presence of "asymptotically uniform clique geometries" on PCCs in a certain range of the parameters.

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