Let Y = ( X, ( R i ), 0⩽ i⩽ d) be a symmetric d-class association scheme, with intersection numbers p ij h and Krein parameters q ij h (0 ⩽ h, i, j ⩽ d), and for each i (0 ⩽ i ⩽ d) define the ith (reduced) intersection diagram D i (resp. representation diagram D i ∗ ) on the nodes 0, 1,…, d by drawing an undirected arc between any distinct h, j for which p ij h > 0 (resp. q ij h > 0). Y is called P-polynomial (resp. Q-polynomial) if some D i (resp. D i ∗ ) is a path. We obtain positive semi-definite matrices G( i) and G(i) ∗ (0 ⩽ i ⩽ d) that yield new inequalities for the p ij h and q ij h . We show for each i (0 ⩽ i ⩽ d), D i ∗ being a forest, the vanishing of G( i), and the existence of a certain geometric representation of X are all equivalent. A similar result relates G(i) ∗ and D i . Denoting by a leaf in any diagram a node adjacent to exactly one other, we show there is at most one leaf besides the O-node in any connected D i ∗ for a P-polynomial scheme. We combine this with the above results and get an interpretation of the Q-polynomial property for P-polynomial schemes. Finally, we use equations induced by the vanishing of some G( i) to obtain a simple proof of a theorem of D. Leonard, that the intersection numbers of a P- and Q-polynomial scheme can be found from 5 parameters.