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Let Γ denote a bipartite distance-regular graph with diameter D > 4, valency k > 3, and intersection numbers ci, bi (0 6 i 6 D). By a pseudo cosine sequence of Γ we mean a sequence of complex scalars σ0, σ1, . . . , σD such that σ0 = 1 and ciσi−1 + biσi+1 = kσ1σi for 1 6 i 6 D − 1. By an associated pseudo primitive idempotent of Γ, we mean a nonzero scalar multiple of the matrix ∑D i=0 σiAi, where A0, A1, . . . , AD are the distance matrices of Γ. Given pseudo primitive idempotents E,F of Γ, we define the pair E,F to be taut whenever the entry-wise product E ◦ F is not a scalar multiple of a pseudo primitive idempotent, but is a linear combination of two pseudo primitive idempotents of Γ. In this paper, we determine all the taut pairs of pseudo primitive idempotents of Γ.

Bipartite distance-regular graphs and taut pairs of pseudo primitive idempotents