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Taut distance-regular graphs of odd diameter. (English) Zbl 1014.05072
A distance-regular graph \(\Gamma\) with diameter \(D\geq 4\), valency \(k\geq 3\) and eigenvalues \(\theta_0>\cdots >\theta_D\) is considered. Let \(M\) denote the Bose-Mesner algebra of \(\Gamma\). For \(0\leq i\leq D\), let \(E_i\) denote the primitive idempotents of \(M\) associated with \(\theta_i\) (\(E_0\) and \(E_d\) are called trivial idempotents). A pair \(E,F\) of nontrivial primitive idempotents of \(M\) is taut whenever the entry-wise product \(E\circ F\) is a linear combination of two primitive idempotents of \(M\). The graph \(\Gamma\) is taut whenever \(\Gamma\) has at least one taut pair of primitive idempotents, but \(\Gamma\) is not 2-homogeneous in the sense of Nomura-Curtin.
In this paper the taut graphs of odd diameter \(D\) are investigated. Let \(E,F\) be nontrivial primitive idempotents of \(M\) and let \(\sigma_0,\dots ,\sigma_d\) and \(\rho_0,\dots ,\rho_d\) denote the cosine sequences of \(E\) and \(F\), respectively. The pair \(E,F\) is taut if and only if there exist real scalars \(\alpha\) and \(\beta\) such that \[ \sigma_{i+1}\rho_{i+1}-\sigma_{i-1}\rho_{i-1}= \alpha\sigma_i(\rho_{i+1}-\rho_{i-1})+\beta\rho_i(\sigma_{i+1}-\sigma_{i-1})\quad (1\leq i\leq D-1). \] If \(D\) is odd and the pair \(E,F\) is taut, then all intersection numbers of \(\Gamma\) are determined in terms of \(\sigma_1,\rho_1,\alpha,\beta\) (Theorem 5.8). As \(E_1,E_d\) is a taut pair in any taut graph for \(d=(D-1)/2\), all intersection numbers of taut graphs \(\Gamma\) are determined in terms of \(k,\mu,\theta_1,\theta_d\) (Corollary 5.9). If \(\Gamma\) is taut and \(D\) is odd, then \(\Gamma\) is an antipodal 2-cover (Theorem 6.4).

MSC:
05E30 Association schemes, strongly regular graphs
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