Ukrainian Mathematical Journal

, Volume 54, Issue 7, pp 1137–1146 | Cite as

On the Nonexistence of Strongly Regular Graphs with Parameters (486, 165, 36, 66)

  • A. A. Makhnev


We prove that a strongly regular graph with parameters (486, 165, 36, 66) does not exist. Since the parameters indicated are parameters of a pseudogeometric graph for pG2(5, 32), we conclude that the partial geometries pG2(5, 32) and pG2(32, 5) do not exist. Finally, a neighborhood of an arbitrary vertex of a pseudogeometric graph for pG3(6, 80) is a pseudogeometric graph for pG2(5, 32) and, therefore, a pseudogeometric graph for the partial geometry pG3(6, 80) [i.e., a strongly regular graph with parameters (1127, 486, 165, 243)] does not exist.


Regular Graph Partial Geometry Arbitrary Vertex Pseudogeometric Graph 
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  1. 1.
    A. E. Brouwer and J. H. van Lint, “Strongly regular graphs and partial geometries,” Kibern. Sb., 24, 186–2229 (1987).Google Scholar
  2. 2.
    A. A. Makhnev, “Locally GQ (4,8) graphs and partial geometries,” Abstracts of the International Algebraic Conference Dedicated to the Memory of A. G. Kurosh (May 24–31, 1998), Moscow (1998), p. 80.Google Scholar
  3. 3.
    S. A. Hobart and D. R. Hughes, “EpGs with minimal μ,” Geom. Dedic., 42, 129–138 (1992).Google Scholar
  4. 4.
    A. A. Makhnev, “On extensions of partial geometries containing small μ-subgraphs,” Diskret. Analiz Issled. Oper., 3, No. 3, 71–83 (1996).Google Scholar
  5. 5.
    A. A. Makhnev, “On pseudogeometric graphs of certain partial geometries,” Probl. Alg. (Gomel), 11, 60–67 (1997).Google Scholar
  6. 6.
    J.-M. Goethals and J. J. Seidel, “The regular graph on 276 points,” Discr. Math., 12, No. 1, 143–158 (1975).Google Scholar
  7. 7.
    A. A. Makhnev, “On strongly regular extensions of generalized quadrangles,” Mat. Sb., 184, No. 12, 123–132 (1993).Google Scholar
  8. 8.
    H. A. Wilbrink and A. E. Brouwer, “(57, 14, 1) strongly regular graph does not exist,” Proc. Kon. Ned. Akad. Wetensch. A., 45, No. 1, 117–121 (1983).Google Scholar

Copyright information

© Plenum Publishing Corporation 2002

Authors and Affiliations

  • A. A. Makhnev
    • 1
  1. 1.Institute of Mathematics and MechanicsRussian Academy of Sciences, Ural DivisionEkaterinburgRussia

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