Taut Distance-Regular Graphs of Odd Diameter

Abstract

Let Γ denote a bipartite distance-regular graph with diameter D ≥ 4, valency k ≥ 3, and distinct eigenvalues t0 > t1 > ··· > tD. Let M denote the Bose-Mesner algebra of Γ. For 0 ≤ i ≤ D, let Ei denote the primitive idempotent of M associated with ti. 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 c 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 σi + 1ρi + 1 − σi − 1ρi − 1 e ασi(ρi + 1 − ρi − 1) + βρi(σi + 1 − σi − 1) (1 ≤ i ≤ D − 1), where σ0, σ1, …, σD and ρ0, ρ1, …, ρD denote the cosine sequences of E, F, respectively. We define Γ to be taut whenever Γ has at least one taut pair of primitive idempotents but Γ is not 2-homogeneous in the sense of Nomura and Curtin. Assume Γ is taut and D is odd, and assume the pair E, F is taut. We show \frac{\sigma_{i+1}-\alpha \sigma_i}{\sigma \sigma_i - \sigma_{i-1}} &=& \frac {\beta \rho_i - \rho_{i-1}}{\rho \rho_i - \rho_{i-1}}, \frac{\rho_{i+1} - \beta \rho_i}{\rho \rho_i - \rho_{i-1}} &=& \frac{\alpha \sigma_i - \sigma_{i-1}}{\sigma \sigma_i - \sigma_{i-1}} for 1 ≤ i ≤ D − 1, where σ e σ1, ρ e ρ1. Using these equations, we recursively obtain σ0, σ1, …, σD and ρ0, ρ1, …, ρD in terms of the four real scalars σ, ρ, α, β. From this we obtain all intersection numbers of Γ in terms of σ, ρ, α, β. We showed in an earlier paper that the pair E1, Ed is taut, where d e (D − 1)/2. Applying our results to this pair, we obtain the intersection numbers of Γ in terms of k, μ, t1, td, where μ denotes the intersection number c2. We show that if Γ is taut and D is odd, then Γ is an antipodal 2-cover.