Abstract
A construction of a pair of strongly regular graphs Гn and Г′n of type L 2n−1(4n−1) from a pair of skew-symmetric association schemes W, W′ of order 4n−1 is presented. Examples of graphs with the same parameters as Гn and Г′n, i.e., of type L 2n−1(4n−1), were known only if 4n−1=p 3, where p is a prime. The first new graph appearing in the series has parameters (v, k, λ)=(225, 98, 45). A 4-vertex condition for relations of a skew-symmetric association scheme (very similar to one for the strongly regular graphs) is introduced and is proved to hold in any case. This has allowed us to check the 4-vertex condition for Гn and Г′n, thus to prove that Гn and Г′n are not rank three graphs if n>2.
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References
Bannai E., Ito T., Algebraic Combinatorics. I. Association Schemes, Benjamin/Cummings, London, 1984.
Brouwer A. E., Preprint, 1987.
Brouwer A. E., van Lint J. H., Strongly regular graphs and partial geometries, In: Enumerations and Designs, Acad. Press, 1984, pp. 85–122.
Delsarte P., An algebraic approach to the association schemes of coding theory, Philips Res. Rep. Suppl., No. 10, 1973.
Faradžev I. A., Ivanov A. A., Klin M. H., Galois correspondence between permutation groups and cellular rings (association schemes), Graphs and Combinatorics 6 (1990), 303–332.
Faradžev I. A., Ivanov A. V., Klin M. H., Pasechnik D. V., Cellular subrings of Cartesian product of cellular rings, In: Proc. Int. Workshop Algebraic and Combinatorial Coding Theory, Informa, Sofia, 1988, pp. 58–62.
Gol'fand Ya. Yu., Klin M. H., Amorphic cellular rings. I, In: Investigations in the Algebraic Theory of Combinatorial Objects, VNIISI, Moscow, 1985, pp. 32–38.
Goethals J. M., Seidel J. J., Orthogonal matrices with zero diagonal, Canad. J. Math. 19 (1967), 1001–1010.
Hall M. Jr, Combinatorial Theory, Blaisdell, Waltham, MA, 1967.
Hestenes M. D., Higman D. G., Rank 3 groups and strongly regular graphs, SIAM-AMS Proc. 4, Providence, 1971, pp. 141–159.
Hubaut X., Strongly regular graphs, Discr. Math. 13 (1975), 357–381.
Ivanov A. V., Amorphic cellular rings. II, In: Investigations in the Algebraic Theory of Combinatorial Objects, VNIISI, Moscow, 1985, pp. 39–49.
Kantor W. M., Liebler R. A., The rank 3 permutation representations of the finite classical groups, Trans. of AMS 271 (1982), 1–71.
Liebeck M. W., Affine permutation groups of rank three, Proc. London Math. Soc. (3) 54 (1987), 477–516.
Ito N., Hadamard tournaments with transitive automorphism groups, Eur. J. Comb. 5 (1984), 37–42.
On Construction and Identification of Graphs, Weisfeiler B. (Ed.), Lect. Notes Math. 558, Springer-Verlag, 1976.
Reid K. B., Brown E., Doubly regular tournaments are equivalent to skew Hadamard matrices, J. Comb. Th. (A) 12 (1972), 332–338.
Szekeres G., Tournaments and Hadamard matrices, Enseignement Math. 15 (1969), 269–278.
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Pasechnik, D.V. Skew-symmetric association schemes with two classes and strongly regular graphs of type L 2n−1(4n−1). Acta Appl Math 29, 129–138 (1992). https://doi.org/10.1007/BF00053382
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DOI: https://doi.org/10.1007/BF00053382