Equisizable partial sum families

Journal of Algebraic Combinatorics volume 51, pages273–296(2020)Cite this article

Abstract

We introduce a special kind of partial sum families, which we call equisizable partial sum families, that can be used to obtain directed strongly regular graphs admitting a semiregular group of automorphisms. We give a construction of an infinite family of equisizable partial sum families depending on two parameters that produce directed strongly regular graphs with new parameters. We also determine the automorphisms group of the associated directed strongly regular graphs in terms of the parameters.

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References

  1. 1.

    Araluze, A., Kutnar, K., Martínez, L., Marušič, D.: Edge connectivity in difference graphs and some new constructions of partial sum families. European J. Combin. 32, 352–360 (2011)

    MathSciNet  Article  Google Scholar 

  2. 2.

    Araluze, A., Kovács, I., Kutnar, K., Martínez, L., Marušič, D.: Partial sum quadruples and bi-Abelian digraphs. J. Combin. Theory Ser. A 119, 1811–1831 (2012)

    MathSciNet  Article  Google Scholar 

  3. 3.

    Bose, R.C.: Strongly regular graphs, partial geometries and partially balanced designs. Pacific J. Math. 13, 389–419 (1963)

    MathSciNet  Article  Google Scholar 

  4. 4.

    de Resmini, M.J., Jungnickel, D.: Strongly regular semi-Cayley graphs. J. Algebraic Combin. 1, 171–195 (1992)

    MathSciNet  Article  Google Scholar 

  5. 5.

    Duval, A.: A directed graph version of strongly regular graphs. J. Combin. Theory Ser. A 47, 71–100 (1988)

    MathSciNet  Article  Google Scholar 

  6. 6.

    Gyürki, Š.: Infinite families of directed strongly regular graphs using equitable partitions. Discrete Math. 339, 2970–2986 (2016)

    MathSciNet  Article  Google Scholar 

  7. 7.

    http://homepages.cwi.nl/~aeb/math/dsrg/dsrg.html

  8. 8.

    Kovács, I., Malnič, A., Marušič, D., Miklavič, Š.: Transitive group actions: (im)primitivity and semiregular subgroups. J. Algebraic Combin. 41, 867–885 (2015)

    MathSciNet  Article  Google Scholar 

  9. 9.

    Kutnar, K., Marušič, D., Miklavič, Š., Šparl, P.: Strongly regular tri-Cayley graphs. European J. Combin. 30, 822–832 (2009)

    MathSciNet  Article  Google Scholar 

  10. 10.

    Malnič, A., Marušič, D., Šparl, P., Frelih, B.: Symmetry structure of bicirculants. Discrete Math. 307, 409–414 (2007)

    MathSciNet  Article  Google Scholar 

  11. 11.

    Martínez, L., Araluze, A.: New tools for the construction of directed strongly regular graphs: difference digraphs and partial sum families. J. Combin. Theory Ser. B 100, 720–728 (2010)

    MathSciNet  Article  Google Scholar 

  12. 12.

    Martínez, L.: Strongly regular m-Cayley circulant graphs and digraphs. Ars Math. Contemp. 8(1), 195–213 (2015)

    MathSciNet  Article  Google Scholar 

  13. 13.

    Milies, C.P., Sehgal, S.K.: An Introduction to Group Rings. Kluwer, Dordrecht (2002)

    Book  Google Scholar 

  14. 14.

    Ramras, M., Donovan, E.: The automorphism group of a Johnson graph. SIAM J. Discrete Math. 25, 267–270 (2011)

    MathSciNet  Article  Google Scholar 

Download references

Acknowledgements

We would like to thank the anonymous referees for valuable comments and suggestions.

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Affiliations

  1. Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA, USA

    Ricardo Grande

  2. FAMNIT, University of Primorska, Glagoljaška 8, 6000, Koper, Slovenia

    István Kovács, Klavdija Kutnar & Dragan Marušič

  3. IMFM, University of Ljubljana, Jadranska 19, 1000, Ljubljana, Slovenia

    Aleksander Malnič & Dragan Marušič

  4. Department of Mathematics, University of the Basque Country UPV/EHU, 48080, Bilbao, Spain

    Luis Martínez

  5. Basque Center of Applied Mathematics (BCAM), 48009, Bilbao, Spain

    Luis Martínez

Authors
  1. Ricardo Grande
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  2. István Kovács
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  3. Klavdija Kutnar
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  4. Aleksander Malnič
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  5. Luis Martínez
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  6. Dragan Marušič
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Corresponding author

Correspondence to Luis Martínez.

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Luis Martínez was supported by the Basque Government Grant IT974-16 and by the UPV/EHU and Basque Center of Applied Mathematics, grant US18/21. The work of István Kovács was supported in part by the Slovenian Research Agency (Research Program P1-0285 and Research Projects N1-0032, N1-0038, N1-0062, J1-6720, J1-7051, J1-9108). The work of Klavdija Kutnar was supported in part by the Slovenian Research Agency (Research Program P1-0285 and Research Projects N1-0032, N1-0038, J1-6720, J1-6743, J1-7051, J1-9110), in part by WoodWisdom-Net\(+\), W\(^3\)B, and in part by NSFC Project 11561021. The work of Dragan Marušič was supported in part by the Slovenian Research Agency (I0-0035, Research Program P1-0285 and Research Projects N1-0032, N1-0038, N1-0062, J1-5433, J1-6720, J1-7051, J1-9108) and in part by H2020 Teaming InnoRenew CoE (Grant No. 739574).

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Grande, R., Kovács, I., Kutnar, K. et al. Equisizable partial sum families. J Algebr Comb 51, 273–296 (2020). https://doi.org/10.1007/s10801-019-00875-w

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Keywords

  • Directed strongly regular graphs
  • Groups of automorphisms
  • Partial sum families