Skip to main content
SpringerLink
Log in

On Automorphisms of a Distance-Regular Graph with Intersection Array {125, 96, 1; 1, 48, 125}

  • Published:
Lobachevskii Journal of Mathematics Aims and scope Submit manuscript

Abstract

J. Koolen posed the problem of studying distance-regular graphs in which the neighborhoods of vertices are strongly regular graphs whose second eigenvalue is at most t for a given positive integer t. This problem is reduced to the description of distance-regular graphs in which the neighborhoods of vertices are strongly regular graphs with a nonprincipal eigenvalue t for t = 1, 2,.... In the paper “Distance regular graphs in which local subgraphs are strongly regular graphs with the second eigenvalue at most 3”, Makhnev and Paduchikh found intersection arrays of distance-regular graphs in which the neighborhoods of vertices are strongly regular graphs with second eigenvalue t, where 2 < t ≤ 3. Graphs with intersection arrays {125, 96, 1; 1, 48, 125}, {176, 150, 1; 1, 25, 176}, and {256, 204, 1; 1, 51, 256} remained unexplored. In this paper, possible orders and fixed-point subgraphs of automorphisms are found for a distance-regular graph with intersection array {125, 96, 1; 1, 48, 125}. It is proved that the neighborhoods of the vertices of this graph are pseudogeometric graphs for GQ(4, 6). Composition factors of the automorphism group for a distance-regular graph with intersection array {125, 96, 1; 1, 48, 125} are determined.

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. A. Makhnev and D.V. Paduchikh, “Remotely regular graphs with strongly regular neighborhoods of vertices and the second eigenvalue not larger than 3,” Dokl. Akad. Nauk 464, 396–400 (2015).

    Google Scholar 

  2. A. E. Brouwer, A.M. Cohen, and A. Neumaier, Distance-Regular Graphs (Springer, New York, 1989).

    Book  MATH  Google Scholar 

  3. P. J. Cameron, Permutation Groups, Vol. 45 of LondonMath. Soc. Student Texts (Cambridge Univ. Press, Cambridge, 1999).

    Book  MATH  Google Scholar 

  4. A. L. Gavrilyuk and A. A. Makhnev, “On automorphisms of a remote graph with an intersection array {56, 45, 1; 1, 9, 56},” Dokl. Akad. Nauk 432, 583–587 (2010).

    MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Khetagurov North Ossetian State University, Vladikavkaz, 362025, Russia

    V. V. Bitkina

  2. Krasovskii Institute of Mathematics and Mechanics, Ural Branch, Russian Academy of Sciences, Yekaterinburg, 620990, Russia

    A. A. Makhnev

Authors
  1. V. V. Bitkina
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. A. A. Makhnev
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to V. V. Bitkina.

Additional information

Original Russian Text © V.V. Bitkina, A.A. Makhnev, 2017, published in Uchenye Zapiski Kazanskogo Universiteta, Seriya Fiziko-Matematicheskie Nauki, 2017, Vol. 159, No. 1, pp. 13–20.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Bitkina, V.V., Makhnev, A.A. On Automorphisms of a Distance-Regular Graph with Intersection Array {125, 96, 1; 1, 48, 125}. Lobachevskii J Math 39, 458–463 (2018). https://doi.org/10.1134/S1995080218030058

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S1995080218030058

Keywords

Search

Navigation