Kite-freeP- andQ-polynomial schemes

Let d denote a non-negative integer. A symmetric, d-class association scheme (or simply a scheme) is a configuration Y = (X, Ri0≤i≤d), where X is a nonempty set and R0, R1, · · · , Rd are non-empty subsets of the Cartesian product X ×X, possessing the following properties. (i) (x, y) ∈ R0 if and only if x = y (x, y ∈ X). (ii) (x, y) ∈ Ri for exactly one i (0 ≤ i ≤ d), (x, y ∈ X). (iii) R i = Ri (0 ≤ i ≤ d), where R i = {(y, x) | (x, y) ∈ Ri} (0 ≤ i ≤ d). (iv) For all integers i, j, k (0 ≤ i, j, k ≤ d), and all x, y ∈ X with (x, y) ∈ Rk, the number pij of z ∈ X such that (x, z) ∈ Ri and (z, y) ∈ Rj is a constant that depends only on i, j, k.