Strongly Regular Graphs Derived from Combinatorial Designs
Canadian journal of mathematics, Tome 22 (1970) no. 3, pp. 597-614

Voir la notice de l'article provenant de la source Cambridge University Press

Several concepts in discrete mathematics such as block designs, Latin squares, Hadamard matrices, tactical configurations, errorcorrecting codes, geometric configurations, finite groups, and graphs are by no means independent. Combinations of these notions may serve the development of any one of them, and sometimes reveal hidden interrelations. In the present paper a central role in this respect is played by the notion of strongly regular graph, the definition of which is recalled below.In § 2, a fibre-type construction for graphs is given which, applied to block designs with λ = 1 and Hadamard matrices, yields strongly regular graphs. The method, although still limited in its applications, may serve further developments. In § 3 we deal with block designs, first considered by Shrikhande [22], in which the number of points in the intersection of any pair of blocks attains only two values.
Goethals, J. M.; Seidel, J. J. Strongly Regular Graphs Derived from Combinatorial Designs. Canadian journal of mathematics, Tome 22 (1970) no. 3, pp. 597-614. doi: 10.4153/CJM-1970-067-9
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[1] 1. Assmus, E. F. and Mattson, H. F., On tactical configurations and error-correcting codes, J. Combinatorial Theory 2 (1967), 243–257. Google Scholar

[2] 2. Bose, R. C., Strongly regular graphs, partial geometries and partially balanced designs, Pacific J. Math. 13 (1963), 389–419. Google Scholar

[3] 3. Ehlich, H., Neue Hadamard-Matrizen, Arch. Math. 16 (1965), 34–36. Google Scholar

[4] 4. Gewirtz, A., Graphs with maximal even girth, Can. J. Math. 21 (1969), 915–934. Google Scholar

[5] 5. Gewirtz, A., The uniqueness ofG(2, 2, 10, 56), Trans. New York Acad. Sci. 81 (1969), 656–675. Google Scholar

[6] 6. Goethals, J. M., On the Golay perfect binary code, J. Combinatorial theory (to appear). Google Scholar

[7] 7. Goethals, J. M. and Seidel, J. J., Orthogonal matrices with zero diagonal, Can. J. Math. 19 (1967), 1001–1010. Google Scholar

[8] 8. Golay, M., Notes on digital coding, IRE Proc. 87 (1949), 637. Google Scholar

[9] 9. Hall, M. Jr., Combinatorial theory (Blaisdell, Waltham, Massachusetts, 1967). Google Scholar

[10] 10. Hall, M. Jr., and Swift, J. D., Determination of Steiner triple systems of order 15, Math. Tables Aids Comput. 9 (1955), 146–156. Google Scholar

[11] 11. Higman, D. G. and Sims, C. C., A simple group of order 44,353,000, Math. Z. 105 (1968), 110–113. Google Scholar

[12] 12. Hughes, D. R., On t-designs and groups, Amer. J. Math. 87 (1965), 761–778. Google Scholar

[13] 13. Karlin, M., New binary coding results by circulants, IEEE Trans. Information Theory 15 (1969), 81–92. Google Scholar

[14] 14. van Lint, J. H. and Seidel, J. J., Equilateral point sets in elliptic geometry, Nederl. Akad. Wetensch. Proc. Ser. A 69 ( = Nederl. Akad. Wetensch. Indag. Math. 28) (1966), 335–348. Google Scholar

[15] 15. Mesner, D. M., A new family of partially balanced incomplete block designs with some Latin square design properties, Ann. Math. Statist. 88 (1967), 571–581. Google Scholar

[16] 16. Paige, L. J., A note on the Mathieu groups, Can. J. Math. 9 (1957), 15–18. Google Scholar

[17] 17. Peterson, W. W., Err or-correcting codes (The M.I.T. Press, Cambridge, 1961.) Google Scholar

[18] 18. Ryser, H. J., Combinatorial mathematics, The Carus Mathematical Monographs, No. 14, Published by Math. Assoc. Amer, ((distributed by) Wiley, New York, 1963). Google Scholar

[19] 19. Seidel, J. J., Strongly regular graphs ofLi-type and of triangular type, Nederl. Akad. Wetensch. Proc. Ser. A 70 (= Nederl. Akad. Wetensch. Indag. Math. 29) (1967), 188–196. Google Scholar

[20] 20. Seidel, J. J., Strongly regular graphs with ( — 1, 1, 0) adjacency matrix having eigenvalue 3, Linear Algebra and Appl. 1 (1968), 281–298. Google Scholar

[21] 21. Seidel, J. J., Strongly regular graphs, Proc. 3rd Waterloo Conference in Combinatorics, pp. 185– 198, Recent Progress in Combinatorics, edited by Tutte, W. T. (Academic Press, New York, 1959). Google Scholar

[22] 22. Shrikhande, S. S., On the dual of some balanced incomplete block designs, Biometrics 8 (1952), 66–72. Google Scholar

[23] 23. Stanton, R. G. and Kalbfleisch, J. G., Quasi-symmetric balanced incomplete block designs, J. Combinatorial Theory 4 (1968), 391–396. Google Scholar

[24] 24. Witt, E., Die 5-fach transitiven Gruppen von Mathieu, Abh. Math. Sem. Hamburg Univ. 12 (1938), 256–264. Google Scholar

[25] 25. Witt, E., Uber Steinersche Système, Abh. Math. Sem. Hamburg Univ. 12 (1938), 265–275. Google Scholar

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