Taut distance-regular graphs of even diameter

Let Γ denote a bipartite distance-regular graph with diameter D⩾4, valency k⩾3, and Bose–Mesner algebra M. Let θ0>θ1>⋯>θD denote the distinct eigenvalues for Γ, and 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 and 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. If Γ is 2-homogeneous in the sense of Nomura and Curtin, then Γ has at least one taut pair of primitive idempotents. We define Γ to be taut whenever Γ has at least one taut pair of primitive idempotents but Γ is not 2-homogeneous. Let θ denote an eigenvalue of Γ other than θ0,θD, and let σ0,σ1,…,σD denote the cosine sequence associated with θ. By a result of Curtin, the following are equivalent: (i) Γ is 2-homogeneous and θ∈{θ1,θD−1}; (ii) there exists a complex scalar λ such that σi−1−λσi+σi+1=0 for 1⩽i⩽D−1. Expanding on this, we show that for D even, the following are equivalent: (i) Γ is taut or 2-homogeneous and θ∈{θ1,θD−1}; (ii) there exists a complex scalar λ such that σi−1−λσi+σi+1=0 for i odd, 1⩽i⩽D−1. Using this result, we show that for D even, Γ is taut or 2-homogeneous if and only if the intersection numbers of Γ are given by certain rational expressions involving D/2 independent variables. We discuss the known examples of taut distance-regular graphs with even diameter.