On the p-Rank of the Adjacency Matrices of Strongly Regular Graphs
A.E. Brouwer
and C.A. van Eijl
DOI: 10.1023/A:1022438616684
Abstract
Let \mathbb F p \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].
Pages: 329–346
Keywords: $p$-rank; strongly regular graphs
Full Text: PDF
References
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.
2. A. Blokhuis and A.R. Calderbank, "Quasi-symmetric designs and the Smith Normal Form," preprint, 1991.
3. A.E. Brouwer, A.M. Cohen, and A. Neumaier, "Distance-regular graphs," Ergebnisse der Mathe- BROUWER AND VAN EIJL matik 3.18, Springer, Heidelberg, 1989.
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.)
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.
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.
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8. C.A. van Eijl, "On the p-rank of the adjacency matrices of strongly regular graphs," MSc. thesis, Techn. Univ. Eindhoven (Oct. 1991).
9. J.-M. Goethals and J.J. Seidel, "The regular two-graph on 276 vertices," Discrete Math. 12 (1975) 143-158.
10. W.H. Haemers, Chr. Parker, V. Pless, and V.D. Tonchev, "A design and a code invariant under the simple group Co3," Report FEW 458, Tilburg Univ., 1990.
11. K. Hoffman and R. Kunze, Linear Algebra, Prentice Hall, 1961.
12. I. Kaplansky, Linear Algebra and Geometry, Chelsea, New York, 1974.
13. E. Lucas, "Sur les congruences des nombres Euleriennes, et des coefficients differentiels des fonctions trigonometriques, suivant un module premier," Bull. Soc. Math. France 6 (1878) 49-54.
14. E. Lucas, Theorie des nombres, Librairie Albert Blanchard, Paris, 1961. (Nouveau tirage.)
15. F.J. MacWilliams and N.J.A. Sloane, The Theory of Error-Correcting Codes, North Holland Publ. Co., Amsterdam, 1977.
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.
17. R.T. Parker, Modular Atlas, preprint (1989).
2. A. Blokhuis and A.R. Calderbank, "Quasi-symmetric designs and the Smith Normal Form," preprint, 1991.
3. A.E. Brouwer, A.M. Cohen, and A. Neumaier, "Distance-regular graphs," Ergebnisse der Mathe- BROUWER AND VAN EIJL matik 3.18, Springer, Heidelberg, 1989.
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.)
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.
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.
7. L.E. Dickson, Theory of Numbers, I, II, III, Chelsea, New York, 1952. (Reprint).
8. C.A. van Eijl, "On the p-rank of the adjacency matrices of strongly regular graphs," MSc. thesis, Techn. Univ. Eindhoven (Oct. 1991).
9. J.-M. Goethals and J.J. Seidel, "The regular two-graph on 276 vertices," Discrete Math. 12 (1975) 143-158.
10. W.H. Haemers, Chr. Parker, V. Pless, and V.D. Tonchev, "A design and a code invariant under the simple group Co3," Report FEW 458, Tilburg Univ., 1990.
11. K. Hoffman and R. Kunze, Linear Algebra, Prentice Hall, 1961.
12. I. Kaplansky, Linear Algebra and Geometry, Chelsea, New York, 1974.
13. E. Lucas, "Sur les congruences des nombres Euleriennes, et des coefficients differentiels des fonctions trigonometriques, suivant un module premier," Bull. Soc. Math. France 6 (1878) 49-54.
14. E. Lucas, Theorie des nombres, Librairie Albert Blanchard, Paris, 1961. (Nouveau tirage.)
15. F.J. MacWilliams and N.J.A. Sloane, The Theory of Error-Correcting Codes, North Holland Publ. Co., Amsterdam, 1977.
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.
17. R.T. Parker, Modular Atlas, preprint (1989).