Non rank 3 strongly regular graphs with the 5-vertex condition


Three new strongly regular graphs on 256, 120, and 135 vertices are described in this paper. They satisfy thet-vertex condition — in the sense of [1] — on the edges and on the nonedges fort=4 but they are not rank 3 graphs. The problem to search for any such graph was discussed on a folklore level several times and was fixed in [2]. Here the graph on 256 vertices satisfies even the 5-vertex condition, and has the graphs on 120 and 135 vertices as its subgraphs. The existence of these graphs was announced in [3] and [4]. [4] contains M. H. Klin's interpretation of the graph on 120 vertices. Further results concerning these graphs were obtained by A. E. Brouwer, cf. [5].

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  1. [1]

    M. D. Hestenes, D. G. Higman, Rank 3 groups and strongly regular graphs.SIAM AMS Proc.,4 (1971), 141–160.

    Google Scholar 

  2. [2]

    M. H.Klin, I. A.Faradjev,V-ring method in permutation group theory and its combinatorial applications (Russian). In “Investigation in applied graph theory” (“Issledovania po prikLladnoi teorii grafov”), Novosibirsk, 1986, 59–97.

  3. [3]

    A. V.Ivanov, Non rank 3 graph with 5-vertex condition. Math. Forschungsinstitut Oberwolfach. Tagungsbericht 24/1987 (Darstellungstheorie endlicher Gruppen), 8–9.

  4. [4]

    A. V.Ivanov and M. H.Klin, The example of strongly regular graph satisfying the 4-vertex conditions which is not rank 3 graph. (Russian) In “Proceeding of XIX All-Union algebraic conference” Part 2, L'vov, 1987, 104.

  5. [5]

    A. E.Brouwer, A. V.Ivanov, M. H.Klin, Some new strongly regular graphs. (To appear).

  6. [6]

    A. E. Brouwer, J. H. van Lint, Strongly regular graphs and partial geometries. In “Enumeration and Design — Proc. Silver Jubilee Conf. on Combinatorics.Waterloo, 1982”, (ed. Jackson D. M., Vanstone S. A.). Acad. Press Toronto, 1984, 85–122.

    Google Scholar 

  7. [7]

    B.Weisfeiler (editor), On the construction and identification of graphs.Lect. Notes Math.,N 558, 1976.

  8. [8]

    I. A. Faradjev, Cellular subrings of symmetric square of rank 3 cellular ring (Russian). In “Investigation in Algebraic theory of combinatorial objects. Proceedings of the seminar.” (“Issledovania po algebraitcheskoi teorii combinatornih obiektov”), Moscow, Institute for System Studies, 1985, p. 76–95.

    Google Scholar 

  9. [9]

    M. H. Klin, Gol'fand Ja. Ju.,Amorphic cellular rings I. (Russian). Ibid, p. 32–38.

    Google Scholar 

  10. [10]

    A. V. Ivanov,Amorphic cellular rings II. (Russian). Ibid, p. 39–49.

    Google Scholar 

  11. [11]

    A. A. Ivanov, I. V. Chuvaeva,Action of group M 12 on Hadamard matrices (Russian). Ibid, p. 159–169.

    Google Scholar 

  12. [12]

    M. H. Klin, One method for construction of primitive graphs (Russian).Scientific works of NKI (Trudi NKI), 1974,N 87, 3–8.

    Google Scholar 

  13. [13]

    R. Mathon, A. Rosa, Tables of parameters of BIBDs withr≦41 including existence, enumeration, and resolvability results. In “Algorithms in combinatorial design theory”,Annals of discrete mathematics 26 (1985), 275–307.

    Google Scholar 

  14. [14]

    M.Hall,Combinatorial Theory. Blaisdell, 1967.

  15. [15]

    E.Bannai, T.Ito, Algebraic Combinatorics I (Association Schemes).The Benjaminommings Publ. Comp., 1984.

  16. [16]

    D. H. Smith, Primitive and imprimitive graphs.Quart. J. Math. (Oxford),22 (1971), 551–557.

    Google Scholar 

  17. [17]

    A. Rudvalis, (Ν, κ, λ)-Graphs and Polarities of (Ν, κ, λ)-Designs.Math. Z.,120 (1971), 224–230.

    Article  Google Scholar 

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Ivanov, A.V. Non rank 3 strongly regular graphs with the 5-vertex condition. Combinatorica 9, 255–260 (1989).

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