On quasi-thin association schemes

TL;DR

This paper characterizes quasi-thin association schemes, distinguishing Kleinian schemes from non-Kleinian ones, and shows that non-Kleinian schemes are essentially two-orbit schemes determined by their intersection numbers.

Contribution

It proves that non-Kleinian quasi-thin schemes are two-orbit schemes and uniquely characterized by their intersection number arrays, expanding understanding of their structure.

Findings

Non-Kleinian quasi-thin schemes are two-orbit schemes.

Non-Kleinian schemes are characterized by their intersection number array.

An infinite family of Kleinian quasi-thin schemes is constructed.

Abstract

An association scheme is called quasi-thin if the valency of each its basic relation is one or two. A quasi-thin scheme is Kleinian if the thin residue of it forms a Klein group with respect to the relation product. It is proved that any Kleinian scheme arises from near-pencil on~$3$ points, or affine or projective plane of order~$2$. The main result is that any non-Kleinian quasi-thin scheme a) is the two-orbit scheme of a suitable permutation group, and b) is characterized up to isomorphism by its intersection number array. An infinite family of Kleinian quasi-thin schemes for which neither a) nor b) holds is also constructed.