A Note on Association Schemes with Two P-Polynomial Structures of Type III

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 th...

Association Schemes with Multiple Q-polynomial Structures

It is well known that an association scheme X = (X, {Ri}0≤i≤d) with k1 > 2 has at most two P polynomial structures. The parametrical condition for an association scheme to have twoP -polynomial structures is also known. In this paper, we give a similar result for Q-polynomial association schemes. In fact, if d > 5, then we obtain exactly the same parametrical conditions for the dual intersectio...

On nonsymmetric P- and Q-polynomial association schemes

If a nonsymmetric P-polynomial association scheme, or equivalently. a distanceregular digraph, has diameter d and girth g. then d = g or d = g 1, by Damerell’s theorem. The dual of this theorem was proved by Leonard. In this paper, we prove that the diameter of a nonsymmetric Pand Q-polynomial association scheme is one less than its girth and its cogirth. We also give a structure theorem for a ...

Some algebra related to P- and Q-polynomial association schemes

Inspired by the theory of P -and Q-polynomial association schemes we consider the following situation in linear algebra. Let F denote a field, and let V denote a vector space over F with finite positive dimension. We consider a pair of linear transformations A : V → V and A : V → V satisfying the following four conditions. (i) A and A are both diagonalizable on V . (ii) There exists an ordering...

A Survey on Omega Polynomial of Some Nano Structures