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Bipartite distance-regular graphs and taut pairs of pseudo primitive idempotents. (English) Zbl 1417.05250
Summary: Let $$\varGamma$$ denote a bipartite distance-regular graph with diameter $$D \ge 4$$, valency $$k \ge 3$$, and intersection numbers $$c_i$$, $$b_i \; (0\le i\le D)$$. By a pseudo cosine sequence of $$\varGamma$$ we mean a sequence of complex scalars $$\sigma _0, \sigma _1, \ldots , \sigma _D$$ such that $$\sigma _0=1$$ and $$c_i \sigma _{i-1} +b_i \sigma _{i+1} =k \sigma _1 \sigma _i$$ for $$1\le i\le{D-1}$$. By an associated pseudo primitive idempotent of $$\varGamma$$, we mean a nonzero scalar multiple of the matrix $$\sum _{i=0}^D \sigma _i A_i$$, where $$A_0, A_1, \ldots , A_D$$ are the distance matrices of $$\varGamma$$. Given pseudo primitive idempotents $$E$$, $$F$$ of $$\varGamma$$, we define the pair $$E, F$$ to be taut whenever the entry-wise product $$E \circ F$$ is not a scalar multiple of a pseudo primitive idempotent, but is a linear combination of two pseudo primitive idempotents of $$\varGamma$$. In this paper, we determine all the taut pairs of pseudo primitive idempotents of $$\varGamma$$.
##### MSC:
 05E30 Association schemes, strongly regular graphs 05C12 Distance in graphs
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