On Quasi-thin Association Schemes with Odd Number of Points

Let (X, R) be an association scheme in the sense of P.-H. Zieschang (1996, “An Algebraic Approach to Association Schemes,” Lecture Notes in Mathematics, Vol. 1628, Springer, New York/Berlin), where X is a finite set and R is a partition of X × X. We say that (X, R) is quasi-thin if each element of R has a valency of at most two. In this paper we focus on quasi-thin association schemes with an odd number of points and obtain that (X, R) has a regular automorphism group when n_O^θ(R) is square-free.