Journal of Algebraic Combinatorics

, Volume 1, Issue 4, pp 329–346 | Cite as

On the p-Rank of the Adjacency Matrices of Strongly Regular Graphs

  • A.E. Brouwer
  • C.A. Van Eijl


Let Γ be a strongly regular graph with adjacency matrix A. Let I be the identity matrix, and J the all-1 matrix. Let p be a prime. Our aim is to study the p-rank (that is, the rank over \(\mathbb{F}_p\), the finite field with p elements) of the matrices M = aA + bJ + cI for integral a, b, c. This note is based on van Eijl [8].

p-rank strongly regular graphs 


  1. 1.
    B. Bagchi, A.E. Brouwer, and H.A. Wilbrink, “Notes on binary codes related to the O(5,q) generalized quadrangle for odd q,” Geom. Dedic. 39 (1991) 339–335.Google Scholar
  2. 2.
    A. Blokhuis and A.R. Calderbank, “Quasi-symmetric designs and the Smith Normal Form,” preprint, 1991.Google Scholar
  3. 3.
    A.E. Brouwer, A.M. Cohen, and A. Neumaier,“Distance-regular graphs,” Ergebnisse der Mathematik 3.18, Springer, Heidelberg, 1989.Google Scholar
  4. 4.
    A.E. Brouwer and W.H. Haemers, “The Gewirtz graph — an exercise in the theory of graph spectra,” Report FEW 486, Tilburg University (April 1991). (To appear in the Vladimir 1991 proceedings.)Google Scholar
  5. 5.
    A.E. Brouwer and J.H. van Lint, “Strongly regular graphs and partial geometries,” pp. 85–122 in Enumeration and Design — Proc. Silver Jubilee Conf. on Combinatorics, Waterloo, 1982 (D.M. Jackson and S.A. Vanstone, eds.), Academic Press, Toronto, 1984. MR 87c:05033; Zbl 555.05016 Russian transl. in Kibern. Sb. Nov. Ser. 24 (1987) 186–229 Zbl 636.05013.Google Scholar
  6. 6.
    J.H. Conway, R.T. Curtis, S.P. Norton, R.P. Parker, and R.A. Wilson, Atlas of Finite Groups, Clarendon Press, Oxford, 1985.Google Scholar
  7. 7.
    L.E. Dickson, Theory of Numbers, I, II, III, Chelsea, New York, 1952. (Reprint).Google Scholar
  8. 8.
    C.A. van Eijl, “On the p-rank of the adjacency matrices of strongly regular graphs,” MSc. thesis, Techn. Univ. Eindhoven (Oct. 1991).Google Scholar
  9. 9.
    J.-M. Goethals and J.J. Seidel, “The regular two-graph on 276 vertices,” Discrete Math. 12 (1975) 143–158.Google Scholar
  10. 10.
    W.H. Haemers, Chr. Parker, V. Pless, and V.D. Tonchev, “A design and a code invariant under the simple group Co 3,” Report FEW 458, Tilburg Univ., 1990.Google Scholar
  11. 11.
    K. Hoffman and R. Kunze, Linear Algebra, Prentice Hall, 1961.Google Scholar
  12. 12.
    I. Kaplansky, Linear Algebra and Geometry, Chelsea, New York, 1974.Google Scholar
  13. 13.
    E. Lucas, “Sur les congruences des nombres Euleriennes, et des coefficients différentiels des fonctions trigonométriques, suivant un module premier,” Bull. Soc. Math. France 6 (1878) 49–54.Google Scholar
  14. 14.
    E. Lucas, Théorie des nombres, Librairie Albert Blanchard, Paris, 1961. (Nouveau tirage.)Google Scholar
  15. 15.
    F.J. MacWilliams and N.J.A. Sloane, The Theory of Error-Correcting Codes, North Holland Publ. Co., Amsterdam, 1977.Google Scholar
  16. 16.
    J. MacWilliams and H.B. Mann, “On the p-rank of the design matrix of a difference set,” Info. and Control 12 (1968) 474–488.Google Scholar
  17. 17.
    R.T. Parker, Modular Atlas, preprint (1989).Google Scholar

Copyright information

© Kluwer Academic Publishers 1992

Authors and Affiliations

  • A.E. Brouwer
    • 1
  • C.A. Van Eijl
    • 1
  1. 1.Dept. of Math.Eindhoven Univ. of TechnologyEindhovenNetherlands

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