论文信息 - On the p-Rank of the Adjacency Matrices of Strongly Regular Graphs

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

AbstractLet 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].

A. Brouwer | C. A. van Eijl

[1] É. Lucas,et al. Sur les congruences des nombres eulériens et des coefficients différentiels des fonctions trigonométriques suivant un module premier , 1878 .

[2] É. Lucas,et al. Théorie des nombres , 1961 .

[3] G. B. Mathews. Theory of numbers , 1963 .

[4] F. Jessie MacWilliams,et al. On the p-Rank of the Design Matrix of a Difference Set , 1968, Inf. Control..

[5] Irving Kaplansky. Linear algebra and geometry , 1969 .

[6] J. J. Seidel,et al. The regular two-graph on 276 vertices , 1975, Discret. Math..

[7] W. Greub. Linear Algebra , 1981 .

[8] Andries E. Brouwer,et al. Strongly regular graphs and partial geometries , 1984 .

[9] J. Conway,et al. ATLAS of Finite Groups , 1985 .

[10] A. Neumaier,et al. Distance Regular Graphs , 1989 .

[11] Andries E. Brouwer,et al. Notes on binary codes related to the O(5, q) generalized quadrangle for odd q , 1991 .

[12] A. Robert Calderbank,et al. Quasi-symmetric designs and the Smith Normal Form , 1992, Des. Codes Cryptogr..

[13] Andries E. Brouwer,et al. The Gewirtz Graph: An Exercise in the Theory of Graph Spectra , 1993, Eur. J. Comb..

[14] Vladimir D. Tonchev,et al. A Design and a Code Invariant under the Simple Group Co3 , 1993, J. Comb. Theory, Ser. A.