Near Hexagons with Two Possible Orders for the Quads

Abstract

We study near hexagons which satisfy the following properties:(i) every two points at distance 2 from each other are contained in a unique quad of order \((s,r_1)\) or \((s,r_2), r_1\neq r_2\); (ii) every line is contained in the same number of quads; (iii) every two opposite points are connected by the same number of geodesics. We show that there exists an association scheme on the point set of such a near hexagon and calculate the intersection numbers. We also show how the eigenvalues of the collinearity matrix and their corresponding multiplicities can be calculated. The fact that all multiplicities and intersection numbers are nonnegative integers gives restrictions on the parameters of the near hexagon. We apply this to the special case in which the near hexagon has big quads.