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

Kivva, Bohdan

doi:10.1016/j.jctb.2021.02.003

Cited by 7 publications

(9 citation statements)

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“…We proved Conjecture 1.3 for the class of metric schemes in [19], [18] (see Theorem 1.15 in this paper). Metric schemes are coherent configurations with edge colors induced by a graph metric, and are essentially equivalent to distance-regular graphs.…”

Section: Theorem 14 (Main)mentioning

confidence: 92%

“…In this paper we studied the problem of classifying primitive coherent configurations with sublinear motion in the case of rank 4. Earlier, in [18,19] Proof. We can assume k ≤ m/2.…”

Section: Summary and Open Questionsmentioning

confidence: 93%

“…Note that this spectral tool gives trivial bound for bipartite graphs, as ξ(X) = k for a k-regular bipartite graph X. A bipartite version of this lemma is established in [19].…”

Section: Spectral Toolmentioning

confidence: 97%

“…In that paper, Babai used a combination of an old combinatorial tool [1] and an introduced spectral tool. We used similar approach in [18,19] to get bounds on the motion of distance-regular graphs of bounded diameter. Definition 1.8.…”

Section: Motion Lower Bounds: the Main Toolsmentioning

confidence: 99%

“…Primitive coherent configurations of rank 4 naturally split into three classes: configurations induced by a primitive distance-regular graph of diameter 3, association schemes of diameter 2 (see Definition 2.10), and primitive coherent configurations with one undirected color and two oriented colors. The case of distance-regular graphs of diameter 3 follows from the results we proved in [18,19]. Define a crown graph to be a regular complete biprtite graph with one perfect matching deleted.…”

Section: Outline Of the Proof Of The Main Theoremmentioning

confidence: 99%

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On the automorphism groups of rank-4 primitive coherent configurations

Kivva¹

2021

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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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Section: Theorem 14 (Main)mentioning

confidence: 92%

“…In this paper we studied the problem of classifying primitive coherent configurations with sublinear motion in the case of rank 4. Earlier, in [18,19] Proof. We can assume k ≤ m/2.…”

Section: Summary and Open Questionsmentioning

confidence: 93%

“…Note that this spectral tool gives trivial bound for bipartite graphs, as ξ(X) = k for a k-regular bipartite graph X. A bipartite version of this lemma is established in [19].…”

Section: Spectral Toolmentioning

confidence: 97%

Section: Motion Lower Bounds: the Main Toolsmentioning

confidence: 99%

Section: Outline Of the Proof Of The Main Theoremmentioning

confidence: 99%

See 3 more Smart Citations