Skip to main content
SpringerLink
Log in

Strongly regular graphs decomposable into a divisible design graph and a Hoffman coclique

  • Published:
Designs, Codes and Cryptography Aims and scope Submit manuscript

Abstract

In 2022, the second author found a prolific construction of strongly regular graphs, which is based on joining a coclique and a divisible design graph with certain parameters. The construction produces strongly regular graphs with the same parameters as the complement of the symplectic graph \(\textsf{Sp}(2d,q)\). In this paper, we determine the parameters of strongly regular graphs which admit a decomposition into a divisible design graph and a coclique attaining the Hoffman bound. In particular, it is shown that when the least eigenvalue of such a strongly regular graph is a prime power, its parameters coincide with those of the complement of \(\textsf{Sp}(2d,q)\). Furthermore, a generalization of the construction is discussed.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Abiad A., Haemers W.H.: Switched symplectic graphs and their 2-ranks. Des. Codes Cryptogr. 81, 35–41 (2016). https://doi.org/10.1007/s10623-015-0127-x.

    Article  MathSciNet  Google Scholar 

  2. Bracken C., McGuire G., Ward H.: New quasi-symmetric designs constructed using mutually orthogonal latin squares and hadamard matrices. Des. Codes Cryptogr. 41(2), 195–198 (2006). https://doi.org/10.1007/s10623-006-9095-6.

    Article  MathSciNet  Google Scholar 

  3. Brouwer A.E., Haemers W.H.: Spectra of Graphs. Springer, New York (2012). https://doi.org/10.1007/978-1-4614-1939-6.

  4. Brouwer A.E., Ihringer F., Kantor W.M.: Strongly regular graphs satisfying the 4-vertex condition (2021). arXiv:2107.00076.

  5. Brouwer A.E., Maldeghem H.V.: Strongly Regular Graphs. Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge (2022). https://doi.org/10.1017/9781009057226.

  6. Cossidente A., Pavese F.: Strongly regular graphs from classical generalized quadrangles. Des. Codes Cryptogr. 85, 457–470 (2017). https://doi.org/10.1007/s10623-016-0318-0.

    Article  MathSciNet  Google Scholar 

  7. Crnković D., Haemers W.H.: Walk-regular divisible design graphs. Des. Codes Cryptogr. 72, 165–175 (2014). https://doi.org/10.1007/s10623-013-9861-0.

    Article  MathSciNet  Google Scholar 

  8. Delsarte P.: An algebraic approach to the association schemes of coding theory. Philips Res. Rep. Suppl. 10 (1973).

  9. Erickson M., Fernando S., Haemers W.H., Hardy D., Hemmeter J.: Deza graphs: a generalization of strongly regular graphs. J. Comb. Des. 7, 359–405 (1999).

    Article  MathSciNet  Google Scholar 

  10. Evans R.J.: AGT, Algebraic Graph Theory. GAP Package (2022). https://gap-packages.github.io/agt.

  11. Fon-Der-Flaass D.G.: New prolific constructions of strongly regular graphs. Adv. Geom. 2, 301–306 (2002).

    MathSciNet  Google Scholar 

  12. Goryainov S., Shalaginov L.V.: Deza graphs: a survey and new results (2021). arXiv:2103.00228.

  13. Haemers W.H.: Eigenvalue techniques in design and graph theory. PhD thesis, Technische Universiteit Eindhoven (1979).

  14. Haemers W.H., Higman D.G.: Strongly regular graphs with strongly regular decomposition. Linear Algebra Appl. 114–115, 379–398 (1989). https://doi.org/10.1016/0024-3795(89)90471-0.

    Article  MathSciNet  Google Scholar 

  15. Haemers W.H., Kharaghani H., Meulenberg M.: Divisible design graphs. J. Comb. Theory Ser. A 118, 978–992 (2011). https://doi.org/10.1016/j.jcta.2010.10.003.

    Article  MathSciNet  Google Scholar 

  16. Hoffman A.J.: On eigenvalues and colorings of graphs. In: Harris, B. (ed.) Graph Theory and Its Applications, pp. 79–91. Academic Press, New York (1970). https://doi.org/10.1142/9789812796936_0041.

  17. Ihringer F.: A switching for all strongly regular collinearity graphs from polar spaces. J. Algebr. Comb. 46, 263–274 (2017). https://doi.org/10.1007/s10801-017-0741-y.

    Article  MathSciNet  Google Scholar 

  18. Kabanov V.V.: New versions of the Wallis-Fon-Der-Flaass construction to create divisible design graphs. Discret. Math. 345, 113054 (2022). https://doi.org/10.1016/j.disc.2022.113054.

    Article  MathSciNet  Google Scholar 

  19. Kabanov V.V.: A new construction of strongly regular graphs with parameters of the complement symplectic graph. Electron. J. Comb. (2023). https://doi.org/10.37236/11343.

    Article  MathSciNet  Google Scholar 

  20. Kantor W.M.: Strongly regular graphs defined by spreads. Isr. J. Math. 41, 298–312 (1982). https://doi.org/10.1007/BF02760536.

    Article  MathSciNet  Google Scholar 

  21. Klin M., Reichard S., Woldar A.: Siamese Combinatorial Objects via Computer Algebra Experimentation, pp. 67–112. Springer, Berlin (2009).

    Google Scholar 

  22. Krcadinac V.: Quasi-symmetric designs. Web-page. https://web.math.pmf.unizg.hr/%7Ekrcko/results/quasisym.html.

  23. Krcadinac V.: Steiner 2-designs. Web-page. https://web.math.pmf.unizg.hr/%7Ekrcko/results/steiner.html.

  24. Krcadinac V., Vlahovic R.: New quasi-symmetric designs by the Kramer-Mesner method. Discret. Math. 339(12), 2884–2890 (2016). https://doi.org/10.1016/j.disc.2016.08.009.

    Article  MathSciNet  Google Scholar 

  25. Kubota S.: Strongly regular graphs with the same parameters as the symplectic graph. Sib. Electron. Math. Rep. 13, 1314–1338 (2016). https://doi.org/10.17377/semi.2016.13.103.

    Article  MathSciNet  Google Scholar 

  26. McDonough T.P., Mavron V.C., Ward H.N.: Amalgams of designs and nets. Bull. Lond. Math. Soc. 41(5), 841–852 (2009). https://doi.org/10.1112/blms/bdp080.

    Article  MathSciNet  Google Scholar 

  27. Muzychuk M.: A generalization of Wallis-Fon-Der-Flaass construction of strongly regular graphs. J. Algebr. Comb. 25, 169–187 (2007). https://doi.org/10.1007/s10801-006-0030-7.

    Article  MathSciNet  Google Scholar 

  28. Neumaier A.: Strongly regular graphs with smallest eigenvalue \(-m\). Arch. Math. (Basel) 33(4), 392–400 (1979–80).

  29. Panasenko D.: Strictly Deza Graphs. Web-Page (2023). http://alg.imm.uran.ru/dezagraphs/main.html.

  30. Seidel J.J.: Strongly regular graphs with \((-1, 1, 0)\) adjacency matrix having eigenvalue 3. Linear Algebra Appl. 1, 281–298 (1968). https://doi.org/10.1016/0024-3795(68)90008-6.

    Article  MathSciNet  Google Scholar 

  31. Spence E.: The strongly regular (40,12,2,4) graphs. Electron. J. Comb. 7(1)(R22) (2000). https://doi.org/10.37236/1500.

  32. Tarry G.: Le problème de 36 officiers. Comptes Rendus de l’Association Française pour l’Avancement des Sciences 1, 122–123 (1900).

    Google Scholar 

  33. Tarry G.: Le problème de 36 officiers. Comptes Rendus de l’Association Française pour l’Avancement des Sciences 2, 170–203 (1901).

    Google Scholar 

  34. van Dam E.R.: Regular graphs with four eigenvalues. Linear Algebra Appl. 226–228, 139–162 (1995). https://doi.org/10.1016/0024-3795(94)00346-F.

    Article  MathSciNet  Google Scholar 

  35. Wallis W.D.: Construction of strongly regular graphs using affine designs. Bull. Aust. Math. Soc. 4, 41–49 (1971). https://doi.org/10.1017/S0004972700046244.

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

We would like to thank the referees for their valuable remarks.The research of Alexander Gavrilyuk is supported by JSPS KAKENHI Grant Number 22K03403.

Author information

Author notes

  1. Alexander L. Gavrilyuk and Vladislav V. Kabanov have contributed equally to this work.

Authors and Affiliations

  1. Shimane University, Nishikawatsu-cho, 1060, Matsue, Shimane, 690-8504, Japan

    Alexander L. Gavrilyuk

  2. Krasovskii Institute of Mathematics and Mechanics, Kovalesvkaya, 16, Yekaterinburg, Russia, 620108

    Vladislav V. Kabanov

Authors
  1. Alexander L. Gavrilyuk
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Vladislav V. Kabanov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Alexander L. Gavrilyuk.

Ethics declarations

Competing interest

Not applicable.

Additional information

Communicated by T. Feng.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Gavrilyuk, A.L., Kabanov, V.V. Strongly regular graphs decomposable into a divisible design graph and a Hoffman coclique. Des. Codes Cryptogr. (2023). https://doi.org/10.1007/s10623-023-01348-9

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s10623-023-01348-9

Keywords

Mathematics Subject Classification

Search

Navigation