On Graphs in Which Neighborhoods of Vertices Are Strongly Regular with Parameters (85,14,3,2) or (325,54,3,10)

Article

Abstract

J. Koolen posed the problem of studying distance-regular graphs in which neighborhoods of vertices are strongly regular graphs with nonprincipal eigenvalue at most t for a given positive integer t. This problem was solved earlier for t = 3. In the case t = 4, the problem was reduced to studying graphs in which neighborhoods of vertices have parameters (352,26,0,2), (352,36,0,4), (243,22,1,2), (729,112,1,20), (204,28,2,4), (232,33,2,5), (676,108,2,20), (85,14,3,2), or (325,54,3,10). In the present paper, we prove that a distance-regular graph in which neighborhoods of vertices are strongly regular with parameters (85, 14, 3, 2) or (325, 54, 3, 10) has intersection array {85, 70, 1; 1, 14, 85} or {325, 270, 1; 1, 54, 325}. In addition, we find possible automorphisms of a graph with intersection array {85, 70, 1; 1, 14, 85}.

Keywords

strongly regular graph locally X-graph automorphism of a graph. 

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Copyright information

© Pleiades Publishing, Ltd. 2017

Authors and Affiliations

  • M. M. Isakova
    • 1
  • A. A. Makhnev
    • 2
    • 3
  • A. A. Tokbaeva
    • 1
  1. 1.Kabardino-Balkarian State UniversityNalchik, Kabardino-Balkar RepublicRussia
  2. 2.Krasovskii Institute of Mathematics and MechanicsUral Branch of the Russian Academy of SciencesYekaterinburgRussia
  3. 3.Ural Federal UniversityYekaterinburgRussia

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