Tight graphs and their primitive idempotents (Q1296388)

Let \(\Gamma\) be a distance-regular graph with diameter \(d\geq 3\) and eigenvalues \(\theta_0>\theta_1>\dots>\theta_d\). Then \[ \Biggl( \theta_1+\frac{k}{a_1+1}\Biggr) \Biggl(\theta_d+\frac{k}{a_1+1}\Biggr)\geq \frac{-ka_1b_1}{(a_1+1)^2}. \] \(\Gamma\) is said to be tight whenever \(\Gamma\) is not bipartite and equality holds above. Suppose \(E\) and \(F\) are nontrivial primitive idempotents of \(\Gamma\) and the entry-wise product \(E\circ F\) is a scalar multiple of a primitive idempotent \(H\) of \(\Gamma\). Then \(\Gamma\) is either bipartite and at least one of \(E,\;F\) is equal to \(E_d\) or tight and \(\{E,F\}=\{E_1,E_d\}\) (in last case the eigenvalue associated with \(H\) is \(\theta_{d-1}\) and \(k\theta_{d-1}= \theta_1\theta_d\)).