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Tight distance-regular graphs and the \(Q\)-polynomial property. (English) Zbl 0993.05147
From the author’s abstract: Let \(\Gamma\) denotes a distance-regular graph with diameter \(d\geq 3\), and assume \(\Gamma\) is tight (in the sense of Jurišić and Terwilliger). Let \(\theta\) denote the second largest or smallest eigenvalue of \(\Gamma\), and let \(\sigma_0,\sigma_1,\dots, \sigma_d\) denote the associated cosine sequence. We obtain an inequality involving \(\sigma_0,\sigma_1,\dots, \sigma_d\) for each integer \(i\) \((1\leq i\leq d-1)\), and we show equality for all \(i\) is closely related to \(\Gamma\) being \(Q\)-polynomial with respect to \(\theta\). We use this idea to investigate the \(Q\)-polynomial structures in tight distance-regular graphs.

05E30 Association schemes, strongly regular graphs
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