On Quasi-thin Association Schemes

An association scheme (or simply, a scheme) is called thin if each of its basic relations has valency 1. It is easy to see that thin schemes can be viewed as groups and, conversely, groups can be seen as thin schemes. In the present paper, we investigate schemes the basic relations of which have valency 1 or 2. We call these schemes quasi-thin. In order to formulate our results we let (X, R) denote a scheme (in the sense of P.-H. Zieschang). We first offer three sufficient conditions for (X, R) to have an automorphism group acting transitively on X. These conditions are (i) Oθ(R)∩Oθ(R)={1}, (ii) nOθ(R)=2, (iii) R possesses an element r such that 〈r〉=R and 〈rr*〉=〈r*r〉. We then prove that, if Oθ(R)=Oθ(R) and nOθ(R)=4, |X|/4∈{3, 4, 7, 8, 12, 16}. As a consequence of the latter result, we obtain a classification of the quasi-thin schemes with |X|=4p, where p is a prime number.