On the automorphism groups of rank-4 primitive coherent configurations

IF 16.4 1区化学 Q1 CHEMISTRY, MULTIDISCIPLINARY

Accounts of Chemical Research Pub Date : 2023-10-11 DOI:10.1016/j.jctb.2023.09.006

Bohdan Kivva

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引用次数: 2

Abstract

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 $\geq c n$ 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.

Recently (March 2022) Sean Eberhard published a class of counterexamples of rank 28 to Babai's conjecture and suggested to replace “Cameron schemes” in the conjecture with a more general class he calls “Cameron sandwiches”. Naturally, our result also confirms the rank 4 case of Eberhard's version of the conjecture.

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关于秩为4的原相干配置的自同构群

置换群G的极小度是不被G的非恒等元固定的最小点数。极小度的下界对G有很强的结构后果。Babai猜想，如果一个有n个顶点的基相干配置不是Cameron方案，那么它的自同构群对于某个常数c>；2014年，Babai证明了强正则图的自同构群的极小度的期望下界，从而证实了秩为3的原始相干配置的猜想。本文将Babai的结果推广到秩为4的原始相干配，证实了这种特殊情况下的猜想。这些证明结合了结构方法和谱方法。最近（2022年3月），Sean Eberhard发表了一类Babai猜想的28阶反例，并建议用他称之为“Cameron三明治”的更一般的类来取代该猜想中的“卡梅伦方案”。当然，我们的结果也证实了Eberhard猜想的秩为4的情况。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Accounts of Chemical Research 化学-化学综合

CiteScore

31.40

自引率

1.10%

发文量

312

审稿时长

2 months

期刊介绍： Accounts of Chemical Research presents short, concise and critical articles offering easy-to-read overviews of basic research and applications in all areas of chemistry and biochemistry. These short reviews focus on research from the author’s own laboratory and are designed to teach the reader about a research project. In addition, Accounts of Chemical Research publishes commentaries that give an informed opinion on a current research problem. Special Issues online are devoted to a single topic of unusual activity and significance. Accounts of Chemical Research replaces the traditional article abstract with an article "Conspectus." These entries synopsize the research affording the reader a closer look at the content and significance of an article. Through this provision of a more detailed description of the article contents, the Conspectus enhances the article's discoverability by search engines and the exposure for the research.