A characterization of skew Hadamard matrices and doubly regular tournaments. (English) Zbl 1251.15033
Skew Hadamard matrices of size \(n\) by \(n\) are characterized in terms of the spectral data of the Seidel matrix of tournaments of order \(n-2\).
Reviewer: Frank Uhlig (Auburn)
MSC:
| 15B34 | Boolean and Hadamard matrices |
| 05B20 | Combinatorial aspects of matrices (incidence, Hadamard, etc.) |
| 05C50 | Graphs and linear algebra (matrices, eigenvalues, etc.) |
| 05C20 | Directed graphs (digraphs), tournaments |
References:
| [1] | (Colbourn, C. J.; Dinits, J. H., The CRC Handbook of Combinatorial Designs (2006), CRC Press: CRC Press Boca Raton) |
| [2] | Cvetković, D., Graphs and their spectra, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz. No., 354-356, 1-50 (1971) · Zbl 0238.05102 |
| [3] | Cvetković; Rowlinson, P., Spectral graph theory, (Beineke, L. W.; Wilson, R. J., Topics in Algebraic Graph Theory (2005), Cambridge University Press: Cambridge University Press Cambridge, UK), 88-112 · Zbl 1059.05071 |
| [4] | Cvetković, D.; Rowlinson, P.; Simić, S., Eigenspaces of graphs, (Encyclopedia of Mathematics and its Applications, vol. 66 (1997), Cambridge University Press: Cambridge University Press Cambridge), xiv+258 pp · Zbl 0878.05057 |
| [5] | Godsil, C.; Royle, G., Algebraic graph theory, (Graduate Texts in Mathematics, vol. 207 (2001), Springer-Verlag: Springer-Verlag New York), xx+439 pp · Zbl 0968.05002 |
| [6] | Horn, R. A.; Johnson, C. R., Matrix Analysis (1990), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0704.15002 |
| [7] | Kirkland, S. J.; Shader, B. L., Tournament matrices with extremal spectral properties, Linear Algebra Appl., 196, 1-17 (1994) · Zbl 0790.15021 |
| [8] | Koukouvinos, C.; Stylianou, S., On skew-Hadamard matrices, Discrete Math., 308, 13, 2723-2731 (2008) · Zbl 1159.05011 |
| [9] | Reid, K. B.; Brown, E., Doubly regular tournaments are equivalent to skew Hadamard matrices, J. Combin. Theory Ser. A, 12, 332-338 (1972) · Zbl 0236.05109 |
| [10] | Rowlinson, P., On 4-cycles and 5-cycles in regular tournaments, Bull. London Math. Soc., 18, 2, 135-139 (1986) · Zbl 0591.05033 |
| [11] | Rowlinson, P., The main eigenvalues of a graph: a survey, Appl. Anal. Discrete Math., 1, 2, 445-471 (2007) · Zbl 1199.05241 |
| [12] | N. Zagaglia Salvi, Some properties of regular tournament matrices (Italian. English summary), in: Proceedings of the Conference on Combinatorial and Incidence Geometry: Principles and Applications (La Mendola, 1982), Rend. Sem. Mat. Brescia, 7, Vita e Pensiero, Milan, 1984, pp. 635-643.; N. Zagaglia Salvi, Some properties of regular tournament matrices (Italian. English summary), in: Proceedings of the Conference on Combinatorial and Incidence Geometry: Principles and Applications (La Mendola, 1982), Rend. Sem. Mat. Brescia, 7, Vita e Pensiero, Milan, 1984, pp. 635-643. · Zbl 0549.05030 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
