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AUTOMORPHISM GROUPS OF BICOSET DIGRAPHS

Part of: Algebraic combinatorics Graph theory

Published online by Cambridge University Press:  24 March 2025

RACHEL BARBER Open the ORCID record for RACHEL BARBER [Opens in a new window] ,
TED DOBSON Open the ORCID record for TED DOBSON [Opens in a new window]  and
GREGORY ROBSON Open the ORCID record for GREGORY ROBSON [Opens in a new window]
RACHEL BARBER
Affiliation:
Department of Mathematics, Hood College, 401 Rosemont Ave., Frederick, MD 21701, USA e-mail: [email protected]
TED DOBSON
Affiliation:
IAM and FAMNIT, University of Primorska, Muzejska trg 2, Koper 6000, Slovenia e-mail: [email protected]
GREGORY ROBSON*
Affiliation:
FAMNIT, University of Primorska, Muzejska trg 2, Koper 6000, Slovenia
*
e-mail: [email protected]
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Abstract

We examine bicoset digraphs and their natural properties from the point of view of symmetry. We then consider connected bicoset digraphs that are X-joins with collections of empty graphs, and show that their automorphism groups can be obtained from their natural irreducible quotients. We further show that such digraphs can be recognised from their connection sets.

Keywords

bicoset graphHaar graphdigraphwreath productX-join

MSC classification

Primary: 05E18: Group actions on combinatorial structures
Secondary: 05C20: Directed graphs (digraphs), tournaments 05C25: Graphs and abstract algebra (groups, rings, fields, etc.) 05C60: Isomorphism problems (reconstruction conjecture, etc.) and homomorphisms (subgraph embedding, etc.) 05C76: Graph operations (line graphs, products, etc.)
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , First View , pp. 1 - 17
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Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

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Footnotes

The work of the second author is supported in part by the Slovenian Research Agency (research programme P1-0285 and research projects N1-0140, N1-0160, J1-2451, N1-0208, J1-3001, J1-3003, J1-4008 and J1-50000), while the work of the third author is supported in part by the Slovenian Research Agency (research program P1-0285 and Young Researchers Grant).

References

Barber, R. and Dobson, T., ‘Finding automorphism groups of double coset graphs and Cayley graphs are equivalent’, Preprint, 2024, arXiv:2407.02316.Google Scholar
Barber, R., Dobson, T. and Robson, G., ‘Automorphism groups of Haar and bicoset digraphs’, Preprint.Google Scholar
Bhoumik, S., Dobson, E. and Morris, J., ‘On the automorphism groups of almost all circulant graphs and digraphs’, Ars Math. Contemp. 7(2) (2014), 487–506.CrossRefGoogle Scholar
Dobson, T., ‘On automorphisms of Haar graphs of abelian groups’, Art Discrete Appl. Math. 5(3) (2022), Article no. 3.06, 22 pages.CrossRefGoogle Scholar
Dobson, T., Malnič, A. and Marušič, D., Symmetry in Graphs, Cambridge Studies in Advanced Mathematics, 198 (Cambridge University Press, Cambridge, 2022).CrossRefGoogle Scholar
Du, J.-L., Feng, Y.-Q. and Spiga, P., ‘On Haar digraphical representations of groups’, Discrete Math. 343(10) (2020), Article no. 112032, 6 pages.Google Scholar
Du, S. and Xu, M., ‘A classification of semisymmetric graphs of order $2 pq$ ’, Comm. Algebra 28(6) (2000), 2685–2715.CrossRefGoogle Scholar
Hemminger, R. L., ‘The group of an $X$ -join of graphs’, J. Combin. Theory 5 (1968), 408–418.CrossRefGoogle Scholar
Hladnik, M., Marušič, D. and Pisanski, T., ‘Cyclic Haar graphs’, Discrete Math. 244(1–3) (2002), 137–152.Google Scholar
Sabidussi, G., ‘Vertex-transitive graphs’, Monatsh. Math. 68 (1964), 426–438.CrossRefGoogle Scholar
Wilson, S., ‘A worthy family of semisymmetric graphs’, Discrete Math. 271(1–3) (2003), 283–294.CrossRefGoogle Scholar