A Note on Thin P-Polynomial and Dual-Thin Q-Polynomial Symmetric Association Schemes

Let Y denote a d-class symmetric association scheme, with d ≥ 3. We show the following: If Y admits a P-polynomial structure with intersection numbers pijh and Y is 1-thin with respect to at least one vertex, then plll=0 → plii=0 1 ≤ i ≤ - 1. If Y admits a Q-polynomial structure with Krein parameters qijh, and Y is dual 1-thin with respect to at least one vertex, then qlll = 0 → qlii = 01 ≤ i ≤ d-1.