On quasi-thin association schemes

An association scheme is called quasi-thin if each of its basic relations has valency 1 or 2. A quasi-thin scheme is called Kleinian if its thin residue is the Klein four-group with respect to relational multiplication. It is proved that any Kleinian quasi-thin scheme arises from a near-pencil on 3 points, from an affine plane of order 2, or from a projective plane of order 2. The main result in this paper is that any non-Kleinian quasi-thin scheme is schurian and separable. We also construct an infinite family of Kleinian quasi-thin schemes which is neither schurian nor separable.