How many P-polynomial Structures can an Association Scheme have?

Abstract

As the Johnson association scheme J (2 k - 1, k - 1) shows, an association scheme sometimes has more than one P -polynomial structure. In this paper we prove that any one association scheme can have no more than two P -polynomial structures (except for the association scheme of an n -gon) and that an association scheme can have two P -polynomial structures in only a few different ways.