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Krein conditions and near polygons. (English) Zbl 0738.05025
Author’s summary: “In this article we present a new proof for (and a generalization of) the Krein condition for association schemes. The proof yields necessary and sufficient conditions for the case of equality. In the special case of regular near polygons we give a second matrix-free proof of the special Krein condition \(q_{dd}^ d\geq 0\) and a corresponding characterization of the equality case. Also, Mathon’s inequality for near hexagons is generalized to arbitrary regular near polygons.”.

05B30 Other designs, configurations
05E30 Association schemes, strongly regular graphs
Full Text: DOI
[1] Biggs, N. L., Algebraic Graph Theory, (Cambridge Tracts in Mathematics, Vol. 67 (1974), Cambridge Univ. Press: Cambridge Univ. Press Cambridge) · Zbl 0501.05039
[2] Biggs, N. L., Automorphic graphs and the Krein condition, Geom. Dedicata, 5, 117-127 (1976) · Zbl 0333.05108
[3] Bose, R. C.; Mesner, D. M., On linear associative algebras corresponding to association schemes of partially balanced designs, Ann. Math. Statist., 30, 21-38 (1959) · Zbl 0089.15002
[4] Brouwer, A. E.; Cohen, A. M.; Neumaier, A., Distance Regular Graphs, (Ergebnisse der Mathematik und ihrer Grenzgebiete, Bd. 18 (1989), Springer: Springer Berlin-Heidelberg), 3. Folge · Zbl 0747.05073
[5] Brouwer, A. E.; Wilbrink, H., The structure of near polygons with quads, Geom. Dedicata, 14, 145-176 (1983) · Zbl 0521.51013
[6] Cameron, P. J.; Goethals, J.-M; Seidel, J. J., Strongly regular graphs having strongly regular subconstituents, J. Algebra, 55, 257-280 (1978) · Zbl 0444.05045
[7] Haemers, W. H.; Roos, C., An inequality for generalized hexagons, Geom. Dedicata, 10, 219-222 (1981) · Zbl 0463.51012
[9] Payne, S.; Thas, J. A., Finite Generalized Quadrangles (1985), Pitman: Pitman New York · Zbl 0551.05027
[10] Scott, L. L., A condition on Higman’s parameters, Notices Amer. Math. Soc., 20, A-97 (1973)
[11] Shult, E. E.; Yanushka, A., Near n-gons and line systems, Geom. Dedicata, 9, 1-72 (1980) · Zbl 0433.51008
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