SEMINAR: Groups and Combinatorics Seminar: Regular near hexagons and Q-polynomial distance-regular graphs

Time and place: 15:00 Friday 24 April in Blakers LT.

Speaker: Bart De Bruyn (Ghent University)

Title: Regular near hexagons and Q-polynomial distance-regular graphs.

Abstract: A near 2d-gon is a point-line geometry with diameter d having the property that for every point x and every line L, there exists a unique point on L nearest to x. A near polygon is called thick if every line is incident with at least three points and regular if its collinearity graph is a so-called distance-regular graph. In my talk, I will discuss thick regular near 2d-gons with a so-called Q-polynomial collinearity graph. For d > 3, we show that apart from Hamming near polygons and dual polar spaces there are no thick Q-polynomial regular near polygons. A thick regular near hexagon is Q-polynomial if and only if t = s^3 + t_2 (s^2 - s + 1), where t + 1 is the number of lines through each point, s + 1 is the number of points on each line and t_2 + 1 is the constant number of common neighbors that two points at distance 2 have. We also show that there cannot exist (necessarily Q-polynomial) regular near hexagons whose parameters (s,t_2,t) are equal to either (3,1,34), (8,4,740), (92,64,1314560), (95,19,1027064) or (105,147,2763012). All these nonexistence results imply the nonexistence of distance-regular graphs with certain parameters. We also mention some applications of these non-existence results.

(Joint work with Frederic Vanhove)