Taut distance-regular graphs of even diameter
Let \(\Gamma\) be a distance-regular graph with diameter \(D\geq 4\) and valency \(k\geq 3\). Let \(k=\theta_0>\cdots >\theta_D\) denote the distinct eigenvalues of \(\Gamma\), and for \(0\leq i\leq D\), let \(E_i\) denote the primitive idempotents of \(\Gamma\) associated with \(\theta_i\). A primitive idempotent \(F\) of \(\Gamma\) is called trivial, if \(F=E_0\) or \(\Gamma\) is bipartite and \(F=E_D\). Let \(E,F\) denote primitive idempotents of \(\Gamma\). The pair \(E,F\) is called taut whenever \(E,F\) are nontrivial and the entry-wise product \(E\circ F\) is a linear combination of two distinct primitive idempotents of \(\Gamma\). We define \(\Gamma\) to be taut, if \(\Gamma\) is not 2-homogeneous and \(\Gamma\) has the taut pair \(E,F\) of primitive idempotents. Theorem. Let \(\Gamma\) be a bipartite distance-regular graph with even diameter \(D\geq 4\), valency \(k\geq 3\), and eigenvalues \(\theta_0>\dots >\theta_D\). Let \(\theta\) denote an eigenvalue of \(\Gamma\) other than \(\theta_0,\theta_D\), and let \(\sigma_0,\dots ,\sigma_D\) denote the cosine sequence associated with \(\theta\). Then the following are equivalent: (i) \(\Gamma\) is taut or 2-homogeneous, and \(\theta\in \{\theta_1,\theta_{D-1}\}\). (ii) There exists a complex number \(\lambda\) such that \(\sigma_{i-1}-\lambda \sigma_i+\sigma_{i+1}=0\) for \(i\) odd, \(1\leq i\leq D-1\). (iii) There exists a complex number \(\lambda\) such that \(\sigma_{i-1}-\lambda \sigma_i+\sigma_{i+1}=0\) for \(i=1,3\).
scientific article; zbMATH DE number 2082036