Taut Distance-Regular Graphs of Odd Diameter

Let Γ denote a bipartite distance-regular graph with diameter D ≥ 4, valency k ≥ 3, and distinct eigenvalues θ0 > θ1 > ··· > θD. Let M denote the Bose-Mesner algebra of Γ. For 0 ≤ i ≤ D, let Ei denote the primitive idempotent of M associated with θi. We refer to E0 and ED as the trivial idempotents of M. Let E, F denote primitive idempotents of M. We say the pair E, F is taut whenever (i) E, F are nontrivial, and (ii) the entry-wise product E ○ F is a linear combination of two distinct primitive idempotents of M. We show the pair E, F is taut if and only if there exist real scalars α, β such that