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Asymptotic enumeration of Cayley digraphs

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Abstract

In this paper we show that almost all Cayley digraphs have automorphism group as small as possible; that is, they are digraphical regular representations (DRRs). More precisely, we show that as r tends to infinity, for every finite group R of order r, out of all possible Cayley digraphs on R the proportion whose automorphism group is as small as possible tends to 1. This proves a natural conjecture first proposed in 1982 by Babai and Godsil.

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Author information

Authors and Affiliations

  1. Department of Mathematics and Computer Science, University of Lethbridge, Lethbridge, AB, T1K 3M4, Canada

    Joy Morris

  2. Dipartimento di Matematica e Applicazioni, University of Milano-Bicocca, Via Cozzi 55, 20125, Milano, Italy

    Pablo Spiga

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  1. Joy Morris
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  2. Pablo Spiga
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Correspondence to Pablo Spiga.

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Supported in part by the National Science and Engineering Research Council of Canada. Address correspondence to P. Spiga

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Morris, J., Spiga, P. Asymptotic enumeration of Cayley digraphs. Isr. J. Math. 242, 401–459 (2021). https://doi.org/10.1007/s11856-021-2150-0

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  • DOI: https://doi.org/10.1007/s11856-021-2150-0

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