Conditions for Singular Incidence Matrices
Willem H. Haemers
Department of Econometrics and O.R. Tilburg University Tilburg The Netherlands
DOI: 10.1007/s10801-005-6907-z
Abstract
Suppose one looks for a square integral matrix N, for which NN T has a prescribed form. Then the Hasse-Minkowski invariants and the determinant of NN T lead to necessary conditions for existence. The Bruck-Ryser-Chowla theorem gives a famous example of such conditions in case N is the incidence matrix of a square block design. This approach fails when N is singular. In this paper it is shown that in some cases conditions can still be obtained if the kernels of N and N T are known, or known to be rationally equivalent. This leads for example to non-existence conditions for self-dual generalised polygons, semi-regular square divisible designs and distance-regular graphs.
Pages: 179–183
Keywords: keywords incidence matrix; bruck-Ryser-chowla theorem; generalised polygon; divisible design; distance-regular graph
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References
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2. A.E. Brouwer, P.J. Cameron, W.H. Haemers, and D.A. Preece, “Self-dual, not self-polar,” Discrete Math., to appear.
3. A.E. Brouwer, A.M. Cohen, and A. Neumaier, Distance-Regular Graphs, Springer, Heidelberg, 1989.
4. M.J. Coster and W.H. Haemers, “Quasi-symmetric designs related to the triangular graph,” Designs, Codes and Cryptography 5 (1995), 27-42.
5. F. De Clerck and H. Van Maldeghem, “Some classes of rank 2 geometries,” in Handbook of Incidence Geometry F. Buekenhout (ed.), Elsevier Science B.V., 1995, pp. 433-475.
6. S.E. Payne and J.A. Thas, “Generalized quadrangles with symmetry, Part I,” Simon Stevin 49 (1975), 3-32.
7. D. Raghavarao, Constructions and Combinatorial Problems in Designs of Experiments, John Wiley & Sons, Inc., 1971.
8. S.S. Shrikhande, D. Raghavarao, and S.K. Tharthare, “Non-existence of some unsymmetrical partially balanced incomplete block designs,” Canad. J. Math. 15 (1963), 686-701.
9. H. Van Maldeghem, Generalized Polygons, Birkh\ddot auser, 1991.