Taut distance-regular graphs of even diameter

Abstract

Let Γ denote a bipartite distance-regular graph with diameter D⩾4, valency k⩾3, and Bose–Mesner algebra M. Let θ 0> θ 1>⋯> θ D denote the distinct eigenvalues for Γ, and for 0⩽ i⩽ D, let E i denote the primitive idempotent of M associated with θ i . We refer to E 0 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 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 θ 0, θ D , and let σ 0, σ 1,…, σ D denote the cosine sequence associated with θ. By a result of Curtin, the following are equivalent: (i) Γ is 2-homogeneous and θ∈{ θ 1, θ D−1 }; (ii) there exists a complex scalar λ such that σ i−1 − λσ i + σ i+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∈{ θ 1, θ D−1 }; (ii) there exists a complex scalar λ such that σ i−1 − λσ i + σ i+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.