Bipartite distance-regular graphs and their primitive idempotents

Let Γ denote a bipartite distance-regular graph with diameter D ≥ 4, valency k ≥ 3, and Bose-Mesner algebra M. Let &thetas;₀ > &thetas;₁ > > &thetas;_D denote the distinct eigenvalues for Γ, and for 0 ≤ i ≤ D , let E_i denote the primitive idempotent of M associated with &thetas;_i. We refer to E₀ and E_D 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 is a linear combination of two distinct primitive idempotents of M. If Γ, is 2-homogeneous in the sense of K. Nomura and B. 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. We prove a number of results concerning the taut condition. Let E, F denote nontrivial primitive idempotents of M, and let &thetas;,&thetas;′ denote the corresponding eigenvalues. We introduce an inequality involving &thetas;,&thetas;′, and we prove that equality holds if and only if the pair E, F is taut. We prove the pair E, F is taut if and only if the cosine sequences of E, F satisfy a certain sequence of equations. Suppose D is odd and Γ is taut. We show the intersection numbers of Γ may be recursively generated from a set of just four parameters. If D is odd and Γ is taut, we show Γ is an antipodal 2-cover. Now suppose D is even. We show 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. If D is even and D ≥ 6, we show Γ is taut or 2-homogeneous if and only if the halved graph of Γ is tight in the sense of Jurišić, Koolen, and Terwilliger.