Skip to main content
SpringerLink
Log in

On Automorphism Groups of AT4(7, 9, r)-Graphs and of Their Local Subgraphs

  • Published:
Proceedings of the Steklov Institute of Mathematics Aims and scope Submit manuscript

Abstract

The paper is devoted to the problem of classification of AT4(p, p + 2, r)-graphs. An example of an AT4(p, p + 2, r)-graph with p = 2 is provided by the Soicher graph with intersection array {56, 45, 16, 1; 1, 8, 45, 56}. The question of existence of AT4(p, p + 2, r)-graphs with p > 2 is still open. One task in their classification is to describe such graphs of small valency. We investigate the automorphism groups of a hypothetical AT4(7, 9, r)-graph and of its local subgraphs. The local subgraphs of each AT4(7, 9, r)-graph are strongly regular with parameters (711, 70, 5, 7). It is unknown whether a strongly regular graph with these parameters exists. We show that the automorphism group of each AT4(7, 9, r)-graph acts intransitively on its arcs. Moreover, we prove that the automorphism group of each strongly regular graph with parameters (711, 70, 5, 7) acts intransitively on its vertices.

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. A. L. Gavrilyuk, A. A. Makhnev, and D. V. Paduchikh, “On distance-regular graphs in which the neighborhoods of vertices are strongly regular,” Dokl. Akad. Nauk 452 (3), 247–251 (2013).

    MathSciNet  MATH  Google Scholar 

  2. A. E. Brouwer, Parameters of Strongly Regular Graphs. http://www.win.tue.nl/~aeb/graphs/srg/srgtab.html

  3. A. E. Brouwer, A. M. Cohen, and A. Neumaier, Distance-Regular Graphs (Springer, Berlin, 1989).

    Book  Google Scholar 

  4. P. J. Cameron, Permutation Groups (Cambridge Univ. Press, Cambridge, 1999).

    Book  Google Scholar 

  5. M. Behbahani and C. Lam, “Strongly regular graphs with nontrivial automorphisms,” Discrete Math. 311 (2–3), 132–144 (2011). doi https://doi.org/10.1016/j.disc.2010.10.005

    Article  MathSciNet  Google Scholar 

  6. A. V. Zavarnitsine, “Finite simple groups with narrow prime spectrum,” Sib. Elektron. Mat. Izv. 6, 1–12 (2009).

    MathSciNet  MATH  Google Scholar 

  7. R. Guralnick, B. Kunyavskiĭ, E. Plotkin, and A. Shalev, “Thompson-like characterizations of the solvable radical,” J. Algebra 300, 363–375 (2006). doi https://doi.org/10.1016/j.jalgebra.2006.03.001

    Article  MathSciNet  Google Scholar 

Download references

Funding

This work was supported by the Russian Science Foundation (project no. 14-11-00061 P).

Author information

Authors and Affiliations

  1. Krasovskii Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences, Yekaterinburg, 620108, Russia

    L. Yu. Tsiovkina

Authors
  1. L. Yu. Tsiovkina
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to L. Yu. Tsiovkina.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Tsiovkina, L.Y. On Automorphism Groups of AT4(7, 9, r)-Graphs and of Their Local Subgraphs. Proc. Steklov Inst. Math. 307 (Suppl 1), 151–158 (2019). https://doi.org/10.1134/S0081543819070125

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0081543819070125

Keywords

Search

Navigation