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CAYLEY GRAPHS AND GRAPHICAL REGULAR REPRESENTATIONS

Part of: Permutation groups Graph theory Algebraic combinatorics

Published online by Cambridge University Press:  15 September 2023

SHASHA ZHENG Open the ORCID record for SHASHA ZHENG [Opens in a new window]
SHASHA ZHENG*
Affiliation:
School of Mathematics and Statistics, The University of Melbourne, Parkville 3010, Victoria, Australia
*
e-mail: zhesz@student.unimelb.edu.au
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Abstract

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Keywords

Cayley graphgraphical regular representationBabai–Godsil–Imrich–Lovász conjecturecubic graphnonabelian simple group

MSC classification

Primary: 05C25: Graphs and abstract algebra (groups, rings, fields, etc.)
Secondary: 05C30: Enumeration in graph theory 05E18: Group actions on combinatorial structures 20B25: Finite automorphism groups of algebraic, geometric, or combinatorial structures
Type
PhD Abstract
Information
Bulletin of the Australian Mathematical Society , Volume 109 , Issue 1 , February 2024 , pp. 161 - 162
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Copyright
© The Author(s), 2023. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

Studying highly symmetric graphs, especially vertex-transitive graphs, is an interesting and challenging topic in algebraic graph theory. In this thesis, we focus our attention on Cayley graphs, a family of vertex-transitive graphs. A graphical regular representation (GRR for short) of a group G is a Cayley graph of G whose full automorphism group is equal to the right regular permutation representation of G.

In 1982, Babai, Godsil, Imrich and Lovász conjectured that except for Cayley graphs of two infinite families of groups—abelian groups of exponent greater than $2$ and generalised dicyclic groups—almost all finite Cayley graphs are GRRs (see [Reference Babai and Godsil1, Conjecture 2.1]). In the study of GRRs of valency three, Xia and Fang [Reference Xia and Fang4] conjectured that with finitely many exceptions, every finite nonabelian simple group has a cubic GRR; and Spiga [Reference Spiga3] conjectured that except for two-dimensional projective special linear groups and a finite number of other cases, every finite nonabelian simple group admits a cubic GRR with connection set containing only one involution. In 1998, Xu [Reference Xu7] introduced the concept of normal Cayley graphs: a Cayley graph of a group is called normal if the right regular permutation representation of the group is normal in the full automorphism group of the graph. In [Reference Xu7], Xu conjectured that except for Cayley graphs of Hamiltonian $2$ -groups, almost all finite Cayley graphs are normal. The purpose of this thesis is to study these problems.

The main work of this thesis consists of two parts, which are presented in Chapter 3 and Chapter 4, respectively. In the first part, we study cubic GRRs of some families of classical groups and, based on some previously known results, confirm the conjecture of Fang and Xia, and a modified version of Spiga’s conjecture. In the second part, we estimate the number of GRRs of a given group with large enough order and confirm the conjecture of Babai, Godsil, Imrich and Lovász as well as the conjecture of Xu.

Some of the research has been published in [Reference Li, Xia, Zhang and Zheng2, Reference Xia, Zheng and Zhou6] or can be found in [Reference Xia and Zheng5].

Footnotes

Thesis submitted to the University of Melbourne in December 2022; degree approved on 15 May 2023; supervisors Binzhou Xia and Sanming Zhou. The thesis was supported by the Melbourne Research Scholarship.

References

Babai, L. and Godsil, C. D., ‘On the automorphism groups of almost all Cayley graphs’, European J. Combin. 3(1) (1982), 9–15.CrossRefGoogle Scholar
Li, J. J., Xia, B., Zhang, X. Q. and Zheng, S., ‘Cubic graphical regular representations of ${\mathrm{PSU}}_3(q)$ ’, Discrete Math. 345(10) (2022), Article no. 112982.CrossRefGoogle Scholar
Spiga, P., ‘Cubic graphical regular representations of finite non-abelian simple groups’, Comm. Algebra 46(6) (2018), 2440–2450.CrossRefGoogle Scholar
Xia, B. and Fang, T., ‘Cubic graphical regular representations of ${\mathrm{PSL}}_2(q)$ ’, Discrete Math. 339(8) (2016), 2051–2055.CrossRefGoogle Scholar
Xia, B. and Zheng, S., ‘Asymptotic enumeration of graphical regular representations’, Preprint, 2023, arXiv:2212.01875.CrossRefGoogle Scholar
Xia, B., Zheng, S. and Zhou, S., ‘Cubic graphical regular representations of some classical simple groups’, J. Algebra 612 (2022), 256–280.CrossRefGoogle Scholar
Xu, M.-Y., ‘Automorphism groups and isomorphisms of Cayley digraphs’, Discrete Math. 182(1–3) (1998), 309–319.CrossRefGoogle Scholar