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The pseudo-cosine sequences of a distance-regular graph. (English) Zbl 1110.05105
Let \(\Gamma \) be a distance-regular graph with diameter \(D\), valency \(k\) and intersection numbers \(a_{i}\), \(b_{i}\), \(c_{i}\). Let \(\sigma _{0}\), \(\sigma _{1}\), \(\dots \), \(\sigma _{D}\) and \(\rho _{0}\), \(\rho _{1}\), \(\dots \), \( \rho _{D}\) denote two pseudo-cosine sequences of \(\Gamma \). This pair of sequences is called tight whenever \(\sigma _{0}\rho _{0}\), \(\sigma _{1}\rho _{1}\), \(\dots \), \(\sigma _{D}\rho _{D}\) is a pseudo-cosine sequence of \( \Gamma \). By definition \(\sigma _{0}\), \(\sigma _{1}\), \(\dots \), \(\sigma _{D} \) is called a pseudo-cosine sequence of \(\Gamma \) if \(\sigma _{0}=1\) and \( c_{i}\sigma _{i-1}+a_{i}\sigma _{i}+b_{1}\sigma _{i+1}=k\sigma _{1}\sigma _{i}\) holds for every \(1\leq i\leq D-1\). The authors find all the tight pairs of pseudo-cosine sequences of \(\Gamma \).

MSC:
05E30 Association schemes, strongly regular graphs
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