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A characterization of $$P$$- and $$Q$$-polynomial association schemes. (English) Zbl 0663.05016
Let $$Y=(X,(R_ i)$$, $$0\leq i\leq d)$$ be a symmetric $$d$$-class association scheme, with intersection numbers $$p^ h_{ij}$$ and Krein parameters $$q^ h_{ij}$$ (0$$\leq h,i,j\leq d)$$, and for each $$i$$ $$(0\leq i\leq d)$$ define the ith (reduced) intersection diagram $$D_ i$$ (resp. representation diagram $$D^*_ i)$$ on the nodes $$0,1,...,d$$ drawing an undirected arc between any distinct $$h$$, $$j$$ for which $$p^ h_{ij}>0$$ (resp. $$q^ h_{ij}>0)$$. $$Y$$ is called $$P$$-polynomial (resp. $$Q$$-polynomial) if some $$D_ i$$ (resp. $$D^*_ i)$$ is a path.We obtain pointwise semi-definite matrices $$G(i)$$ and $$G(i)^*$$ (0$$\leq i\leq d)$$ that yield new inequalities for the $$p^ h_{ij}$$ and $$q^ h_{ij}$$. We show for each $$i$$ $$(0\leq i\leq d)$$, $$D^*_ i$$ 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.

##### MSC:
 05B30 Other designs, configurations
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##### References:
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