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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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