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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.”.

MSC:
 05B30 Other designs, configurations 05E30 Association schemes, strongly regular graphs
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References:
 [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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