Abstract
Divisible design graphs were introduced in 2011 by Haemers, Kharaghani and Meulenberg. Further, divisible design graphs which can be obtained as Cayley graphs were recently studied by Kabanov and Shalaginov. In this paper we give new constructions of divisible design Cayley graphs and classify divisible design Cayley graphs on \(v \le 27\) vertices.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Data Availability
All data generated or analysed during this study are included in this published article.
References
Beth, T., Jungnickel, D., Lenz, H.: Design Theory, 2nd edn. Cambridge University Press, Cambridge (1999)
Bosma, W., Cannon, J. Handbook of Magma Functions, Department of Mathematics, University of Sydney (1994). http://magma.maths.usyd.edu.au/magma
Conway, J.H., Curtis, R.T., Norton, S.P., Parker, R.A., Wilson, R.A., Thackray, J.G.: Atlas of Finite Groups. Clarendon Press, Oxford (1985)
Crnković, D., Haemers, W.H.: Walk-regular divisible design graphs. Des. Codes Cryptogr. 72, 165–175 (2014)
Crnković, D., Kharaghani, H., Švob, A.: Divisible design Cayley digraphs. Discret. Math. 343(111784), 8 (2020)
Diestel, R.: Graph Theory, 5th edn. Springer, Berlin (2017)
Haemers, W.H., Kharaghani, H., Meulenberg, M.: Divisible design graphs. J. Combin. Theory Ser. A 118, 978–992 (2011)
Kabanov, V.V., Shalaginov, L.: On divisible design Cayley graphs. Art Discret. Appl. Math. 4, 9 (2021)
Panasenko, D., Shalaginov, L. Classification of divisible design graphs with at most 39 vertices. arXiv:2109.12805 (preprint)
Robinson, D.: A Course in the Theory of Groups. Springer-Verlag, New York (1996)
Rudvalis, A.: \((v, k, \lambda )\)-graphs and polarities of \((v, k, \lambda )\)-designs. Math. Z. 120, 224–230 (1971)
Sabidussi, G.: On a class of fixed-point-free graphs. Proc. Am. Math. Soc. 9, 800–804 (1958)
Soicher, L.H. GRAPE, GRaph Algorithms using PErmutation groups, Version 4.8.4 (2021) (Refereed GAP package). https://gap-packages.github.io/grape
Acknowledgements
The authors would like to thank the anonymous referee for helpful comments that improved the presentation of the paper.
Funding
This work has been fully supported by Croatian Science Foundation under the projects 6732 and 5713.
Author information
Authors and Affiliations
Contributions
This is a joint collaboration with both authors contributing substantially throughout.
Corresponding author
Ethics declarations
Conflict of interest
The authors have no relevant financial or non-financial interests to disclose.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Crnković, D., Švob, A. New Constructions of Divisible Design Cayley Graphs. Graphs and Combinatorics 38, 17 (2022). https://doi.org/10.1007/s00373-021-02440-4
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/s00373-021-02440-4