A characterization of Johnson and Hamming graphs and proof of Babai's conjecture

Kivva, Bohdan

doi:10.1016/j.jctb.2021.07.003

Cited by 6 publications

(9 citation statements)

References 24 publications

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“…In [16] we confirm the above conjecture in the very special case, when the second largest eigenvalue is sufficiently close to b 1 , and some distance i is dominant. We note that all families mentioned in Conjecture 1.10 have dominant distance d. Among these families, only the Hamming and the Johnson graphs satisfy the assumption that the second largest eigenvalue is close to b 1 .…”

Section: Graphs With Bounded Smallest Eigenvaluesupporting

confidence: 84%

“…In the present paper we cover the case when X is not geometric. Moreover, we prove an eigenvalue lower bound, which will be one of the main ingredients in the analysis of the remaining cases in the companion paper [16]. We confirm Conjecture 1.5 for the cases not covered by the theorem above, i.e., for geometric distance-regular graphs with bounded smallest eigenvalue, in [16].…”

Section: Main Results: Minimal Degree Of the Automorphism Groupsupporting

confidence: 64%

“…The proof of the conjecture is split between two papers 1 due to the different nature of techniques and results obtained. In particular, in [16] we obtain a classification of geometric distance-regular graphs with bounded smallest eigenvalue under certain combinatorial and spectral assumptions, but with no assumption on the motion.…”

Section: Main Results: Minimal Degree Of the Automorphism Groupmentioning

confidence: 99%

“…Remark 1.9. We will give a proof of Conjecture 1.5 in [16], so the theorem above will turn into an unconditional result.…”

Section: Main Results: Minimal Degree Of the Automorphism Groupmentioning

confidence: 99%

“…The new results obtained since that posting warrant the current reorganization of the material. In particular, that preprint did not address the imprimitive case, and none of the main results of the companion paper[16] were available.…”

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confidence: 99%

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On the spectral gap and the automorphism group of distance-regular graphs

Kivva

2019

Preprint

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We prove that a distance-regular graph with a dominant distance is a spectral expander. The key ingredient of the proof is a new inequality on the intersection numbers. We use the spectral gap bound to study the structure of the automorphism group.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. In 2014 Babai proved that the automorphism group of a strongly regular graph with n vertices has minimal degree ≥ cn, with known exceptions. Strongly regular graphs correspond to distance-regular graphs of diameter 2. Babai conjectured that Hamming and Johnson graphs are the only primitive distance-regular graphs of diameter d ≥ 3 whose automorphism group has sublinear minimal degree. We confirm this conjecture for non-geometric primitive distance-regular graphs of bounded diameter. We also show if the primitivity assumption is removed, then only one additional family of exceptions arises, the cocktail-party graphs. We settle the geometric case in a companion paper.

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Section: Graphs With Bounded Smallest Eigenvaluesupporting

confidence: 84%

Section: Main Results: Minimal Degree Of the Automorphism Groupsupporting

confidence: 64%

Section: Main Results: Minimal Degree Of the Automorphism Groupmentioning

confidence: 99%

“…Remark 1.9. We will give a proof of Conjecture 1.5 in [16], so the theorem above will turn into an unconditional result.…”

Section: Main Results: Minimal Degree Of the Automorphism Groupmentioning

confidence: 99%

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confidence: 99%

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On the spectral gap and the automorphism group of distance-regular graphs

Kivva

2019

Preprint

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Hamming Sandwiches

Eberhard¹

2023

Combinatorica

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We describe primitive association schemes $${\mathfrak {X}}$$ X of degree n such that $$\textrm{Aut}({\mathfrak {X}})$$ Aut ( X ) is imprimitive and $$|\textrm{Aut}({\mathfrak {X}})| \ge \exp (n^{1/8})$$ | Aut ( X ) | ≥ exp ( n 1 / 8 ) , contradicting a conjecture of Babai. This and other examples we give are the first known examples of nonschurian primitive coherent configurations (PCC) with more than a quasipolynomial number of automorphisms. Our constructions are “Hamming sandwiches”, association schemes sandwiched between the dth tensor power of the trivial scheme and the d-dimensional Hamming scheme. We study Hamming sandwiches in general, and exhaustively for $$d \le 8$$ d ≤ 8 . We revise Babai’s conjecture by suggesting that any PCC with more than a quasipolynomial number of automorphisms must be an association scheme sandwiched between a tensor power of a Johnson scheme and the corresponding full Cameron scheme. If true, it follows that any nonschurian PCC has at most $$\exp O(n^{1/8} \log n)$$ exp O ( n 1 / 8 log n ) automorphisms.

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