On nonsymmetric P- and Q-polynomial association schemes

If a nonsymmetric P-polynomial association scheme, or equivalently, a distance-regular 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 P- and Q-polynomial association scheme is one less than its girth and its cogirth. We also give a structure theorem for a nonsymmetric Q-polynomial association scheme whose diameter is equal to its cogirth. We use self-duality and unimodality to show that the eigenvalues of a nontrivial nonsymmetric P- and Q-polynomial association scheme are quadratic over the rationals. The fact that the adjacency algebra becomes a C-algebra gives a necessary condition for the existence of a nonsymmetric P- and Q-polynomial association scheme. As an application, it is shown that the only nontrivial nonsymmetric P- and Q-polynomial association scheme with girth 5 is the directed 5 cycle.