A new inequality for bipartite distance-regular graphs

Let Γ denote a bipartite distance-regular graph with diameter D⩾4 and valency k⩾3. Let θ denote an eigenvalue of Γ other than k and −k and consider the associated cosine sequence, σ0,σ1,…,σD. We show(σ1−σi+1)(σ1−σi−1)⩾(σ2−σi)(σ0−σi)for 1⩽i⩽D−1. We show the following are equivalent: (i) equality is attained above for i=3, (ii) equality is attained above for 1⩽i⩽D−1, (iii) there exists a real scalar β such that σi−1−βσi+σi+1 is independent of i for 1⩽i⩽D−1. We say θ is three-term recurrent (or TTR) whenever (i)–(iii) are satisfied. cuss the connection between TTR eigenvalues and the Q-polynomial property. When an eigenvalue is TTR, we find formulae for the intersection numbers and eigenvalues of Γ in terms of at most two free parameters, classifying Γ if β=±2. Among the eigenvalues of Γ in their natural order, we consider which can be TTR. We show Γ can have at most three distinct TTR eigenvalues. We show Γ has three distinct TTR eigenvalues if and only if Γ is 2-homogeneous in the sense of Curtin and Nomura. We show Γ has exactly two distinct TTR eigenvalues if and only if Γ is antipodal with diameter 5, but not 2-homogeneous.