Let denote a bipartite distance-regular graph with diameter D ≥ 4 and valency k ≥ 3. Let θ 0 > θ 1 > · · · > θ D denote the eigenvalues of and let E 0 , E 1 , . . . , E D denote the associated primitive idempotents. Fix s (1 ≤ s ≤ D − 1) and abbreviate E := E s . We say E is a tail whenever the entrywise product E • E is a linear combination of E 0 , E and at most one other primitive idempotent of . Let q h i j (0 ≤ h, i, j ≤ D) denote the Krein parameters of and let denote the undirected graph with vertices 0, 1, . . . , D where two vertices i, j are adjacent whenever i = j and q s i j = 0. We show E is a tail if and only if one of (i)-(iii) holds: (i) is a path; (ii) has two connected components, each of which is a path; (iii) D = 6 and has two connected components, one of which is a path on four vertices and the other of which is a clique on three vertices.
Let Γ denote a Q-polynomial distance-regular graph with diameter D ≥ 4. Assume that the intersection numbers of Γ satisfy a i = 0 for 0 ≤ i ≤ D − 1 and a D = 0. We show that Γ is a polygon, a folded cube, or an Odd graph.
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