Tails of Bipartite Distance-regular Graphs

Let Γ denote a bipartite distance-regular graph with diameterD ≥ 4 and valency k ≥ 3. Let θ 0 > θ 1 > ⋯ > θD denote the eigenvalues of Γ and let E0, E1,⋯ , EDdenote the associated primitive idempotents. Fix s(1 ≤ s ≤ D − 1 ) and abbreviate E: = Es. We say E is a tail whenever the entrywise product E ∘ E is a linear combination of E0, E and at most one other primitive idempotent of Γ. Letqijσi + 1 h (0 ≤ h , i, j ≤ D) denote the Krein parameters of Γ and letΔ denote the undirected graph with vertices 0, 1,⋯ , D where two vertices i, j are adjacent whenever i ≠ = j andqijσi + 1s ≠ = 0. We show E is a tail if and only if one of (i)–(iii) holds: (i) Δ is a path; (ii) Δ has two connected components, each of which is a path; (iii) D = 6 and Δ has two connected components, one of which is a path on four vertices and the other of which is a clique on three vertices.