Bipartite distance-regular graphs and the Q-polynomial property; the combinatorial meaning of q

These problems are inspired by a careful study of the papers of concerning bipartite distance-regular graphs. Throughout these notes we let Γ = (X, R) denote a bipartite distance-regular graph with diameter D ≥ 3 and standard module V = C X. We fix a vertex x ∈ X and let E denote the corresponding dual primitive idempotents. We define the matrices R = D i=0 E * i+1 AE * i , L= D i=0 E * i−1 AE * i , where E * D+1 = 0 and E * −1 = 0. We call R (resp. L) the raising matrix (resp. lowering matrix) with respect to x. We recall R + L = A and R t = L, where t denotes transpose. 1 Motivation: some comments on the Q-polynomial property Let E denote a nontrivial primitive idempotent of Γ and let θ * 0 , θ * 1 ,. .. , θ * D denote the corresponding dual eigenvalue sequence. Throughout this section we assume that Γ is Q-polynomial with respect to E. By Leonard's theorem the expressions θ * i−2 − θ * i+1 θ * i−1 − θ * i (1) are independent of i for 2 ≤ i ≤ D − 1. We define β ∈ R so that β + 1 is equal to the common value of (1). We let q denote a nonzero scalar in C such that q + q −1 = β. Our goal in this section is to get a combinatorial description of q.