A characterization of Q-polynomial distance-regular graphs

Let ΓΓ denote a distance-regular graph with diameter D⩾3D⩾3. Let θθ denote a nontrivial eigenvalue of ΓΓ and let θ0*,θ1*,…,θD* denote the corresponding dual eigenvalue sequence. In this paper we prove that ΓΓ is Q  -polynomial with respect to θθ if and only if the following (i)–(iii) hold:(i)There exist β,γ*∈Cβ,γ*∈C such thatequation(1)γ*=θi-1*-βθi*+θi+1*(1⩽i⩽D-1).(ii)There exist γ,ω,η*∈Cγ,ω,η*∈C such that the intersection numbers aiai satisfyai(θi*-θi-1*)(θi*-θi+1*)=γθi*2+ωθi*+η*for 0⩽i⩽D0⩽i⩽D, where θ-1* and θD+1* are the scalars which satisfy Eq. (1) for i=0i=0, i=Di=D, respectively.(iii)θi*≠θ0* for 1⩽i⩽D1⩽i⩽D.