An inequality involving two eigenvalues of a bipartite distance-regular graph

Abstract

Let Γ denote a bipartite distance-regular graph with diameter D⩾4 and valency k⩾3. Let θ, θ′ denote eigenvalues of Γ other than k and − k. We obtain an inequality involving θ, θ′ and the intersection numbers of Γ, which we refer to as the bipartite fundamental bound (BFB). Let E, F denote the primitive idempotents of Γ associated with θ, θ′, respectively. We show that the following are equivalent: (i) θ, θ′ satisfy equality in BFB; (ii) the entry-wise product E∘ F is a linear combination of at most two primitive idempotents of Γ; (iii) E∘ F is a linear combination of exactly two primitive idempotents of Γ. Let Φ denote the set of pairs θ, θ′, where θ and θ′ are eigenvalues of Γ other than k and − k that satisfy equality in BFB. We determine Φ. The answer depends on a certain expression Δ involving the intersection numbers of Γ. We show Φ≠∅ and Δ=0 if and only if Γ is 2-homogeneous in the sense of Curtin and Nomura. We show that if D is even and at least 6, then Φ≠∅ if and only if the halved graph 1 2 Γ is tight in the sense of Juris̆ić, Koolen, and Terwilliger. We show that for D=4 or D=5, Φ≠∅ if and only if Γ is antipodal.