A Remark on Association Schemes with Two P-polynomial Structures

Gardiner [5] and Bannai and Bannai [1] classified symmetric association schemes (X, {Ro, ... , Rd}) such that for (at least) two different choices of i, the graph (X, Ri ) is distance-regular with diameter d. It turns out that except in the case of a polygon there are never more than two choices for i. Let one of the choices be i = 1, and let the ordering of the relations be such that R j is the relation of being at distance j in the graph r = (X, R 1) (0 ~j ~ d). Then for the other choice we have i E {2, d 1, d}, and the ordering of the relations according to distance in the graph ..1 = (X, R;) is one of the following (cf. Bannai and Ito [2], Theorem III.4.2, p. 241): (I) Ro, R2 , R4 , R6 , ••• , Rs, R3 , R1; (II) Ro, Rd, R 1 , Rdl , R2 , Rd2 , R3 , ••• ;