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An inequality on the cosines of a tight distance-regular graph. (English) Zbl 0979.05112
The author considers a distance-regular graph \(\Gamma\) with diameter \(d\geq 3\), valency \(k\), and eigenvalues \(k=\theta_0>\theta_1>\cdots>\theta_d\), which is tight in the sense of A. Jurišić, J. Koolen, and P. Terwilliger [J. Algebr. Comb. 12, No. 2, 163-197 (2000; Zbl 0959.05121)]. He obtains an inequality involving the first, second and third cosines associated with \(\theta\), when \(\theta\) is \(\theta_1\) or \(\theta_d.\) (\(\theta_1\) and \(\theta_d\) are involved in the definition of a tight graph.) Also, the author proves that equality is attained if and only if \(\Gamma\) is dual biparite \(\mathcal Q\)-polynomial with respect to \(\theta\).

05E30 Association schemes, strongly regular graphs
Full Text: DOI
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