ICM 1998, Berlin, Aug. 18–27

Abstracts of Short Communications and Posters

Section:

13. Combinatorics

1991 MS Classification: 20, 05 Makhnev, Alexandre, Institute of Mathematics and Mechanics, Ekaterinburg, Russia Partial geometries and the Krein condition If S is a geomerty of points and lines, such that every line has s + 1 points, every point is on t + 1 lines (with s > 0, t > 0) and for any antiflag (a, l) there is exactly α points on l, which collinear with a, then S is called a α-partial geometry of order (s, t), or pG_{α}(s, t). A point graph of partial geometry pG_{α}(s, t) is strongly regular with parameters v = (s+1)(1+st/α), k = s(t + 1), λ = (s − 1) + (α − 1)t, μ = α(t + 1). A strongly regular graph Γ with this parameters is called pseudogeometric graph for this geometry. It was proved in [1] that a strongly regular locally GQ(4, 8) graph Γ has parameters (870,165,36,30) or (486,165,36,66). In last case Γ is pseudogeometric graph for partial geometry pG_{2}(5, 32). This graph has equality in the Krein condition: (s + 1 − 2α)t ≤ (s − 1)(s + 1 − α)^{2}. Theorem. A pseudogeometric graph for partial geometry pG_{2}(5, 32) does not exist. This paper supported by the Russion Foundation for Fundamental Investigations (Grant 96-01-00488) Reference: [1] A. A. Makhnev. On strongly regular extensions of generalized quadrangles, Matem. Sbornik 1993, v. 184, 123–132.