On association schemes generated by a relation or an idempotent

In this paper we study the polynomiality and co-polynomiality of association schemes. These two notions were introduced by Ito. We show that P-polynomial association schemes are co-polynomial and Q-polynomial association schemes are polynomial. We also determine the polynomiality and co-polynomiality for 3-class association schemes. Further we give examples of association schemes that are polynomial but not co-polynomial, or co-polynomial but not polynomial.(c) 2023 Elsevier Inc. All rights reserved.