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Pseudo primitive idempotents and almost 2-homogeneous bipartite distance-regular graphs. (English) Zbl 1133.05105
The pseudo cosine sequence of a distance-regular graph $$\Gamma$$ of diameter $$D$$ is a sequence $$\sigma$$ of scalars $$1=\sigma_0,\sigma_1,\dots,\sigma_D$$ defined by a particular recurrence relation involving the intersection numbers of $$\Gamma$$. The pseudo primitive idempotent $$E$$ associated to $$\sigma$$ is a non-0 scalar multiple of the matrix $$E=\sum_i \sigma_i A_i$$, with $$A_i$$’s being the distance matrices of $$\Gamma$$. It is shown that for a bipartite $$\Gamma$$ of valency at least 3 and with $$D\geq 4$$, and for $$\sigma$$ satisfying $$| \sigma_1| \neq 1$$ the following properties are equivalent: (i) $$E\circ E$$ is a linear combination of all-ones matrix and a pseudo primitive idempotent; (ii) there exists $$\beta$$ such that $$\beta\sigma_i=\sigma_{i-1}+\sigma_{i+1}$$ for $$1\leq i \leq D-1$$. Moreover, such $$\sigma$$ and $$E$$ exist if and only if $$\Gamma$$ satisfies $$c_2\geq 2$$ and is almost $$2$$-homogeneous.

##### MSC:
 05E30 Association schemes, strongly regular graphs 05C75 Structural characterization of families of graphs 05C62 Graph representations (geometric and intersection representations, etc.)
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