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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.

##### MSC:
 5e+30 Association schemes, strongly regular graphs
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