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On the primitive idempotents of distance-regular graphs. (English) Zbl 0993.05148
From the author’s abstract: Let $$\Gamma$$ denote a distance-regular graph with diameter $$d\geq 3$$. Let $$E$$, $$F$$ denote nontrivial primitive idempotents of $$\Gamma$$ such that $$F$$ corresponds to the second largest or least eigenvalue. We investigate the situation that there exists a primitive idempotent $$H$$ of $$\Gamma$$ such that $$E\circ F$$ is a linear combination of $$F$$ and $$H$$. Our main purpose is to obtain inequalities involving the cosines of $$E$$, and to show that equality is closely related to $$\Gamma$$ being $$Q$$-polynomial with respect to $$E$$. This generalizes a result of Lang on bipartite graphs and a result of Pascasio on tight graphs.

##### MSC:
 05E30 Association schemes, strongly regular graphs 05C12 Distance in graphs 05C50 Graphs and linear algebra (matrices, eigenvalues, etc.)
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