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On bipartite distance-regular graphs with exactly two irreducible T-modules with endpoint two. (English) Zbl 1352.05196
Summary: Let $$\Gamma$$ denote a bipartite distance-regular graph with diameter $$D \geq 4$$ and valency $$k \geq 3$$. Let $$X$$ denote the vertex set of {$$\Gamma$$}, and let $$A$$ denote the adjacency matrix of $$\Gamma$$. For $$x \in X$$ let $$T = T(x)$$ denote the subalgebra of $$\text{Mat}_X(\mathbb{C})$$ generated by $$A$$, $$E_0^\ast, E_1^\ast, \ldots, E_D^\ast$$, where for $$0 \leq i \leq D$$, $$E_i^\ast$$ represents the projection onto the $$i$$th subconstituent of $$\Gamma$$ with respect to $$x$$. We refer to $$T$$ as the Terwilliger algebra of $$\Gamma$$ with respect to $$x$$. An irreducible $$T$$-module $$W$$ is said to be thin whenever dim $$E_i^\ast W \leq 1$$ for $$0 \leq i \leq D$$. By the endpoint of $$W$$ we mean $$\min\{i \mid E_i^\ast W \neq 0 \}$$. For $$0 \leq i \leq D$$, let $$\operatorname{\Gamma}_i(z)$$ denote the set of vertices in $$X$$ that are distance $$i$$ from vertex $$z$$. Define a parameter $$\operatorname{\Delta}_2$$ in terms of the intersection numbers by $$\operatorname{\Delta}_2 = (k - 2)(c_3 - 1) -(c_2 - 1) p_{22}^2$$. In this paper we prove the following are equivalent: (i) $$\operatorname{\Delta}_2 > 0$$ and for $$2 \leq i \leq D - 2$$ there exist complex scalars $$\alpha_i, \beta_i$$ with the following property: for all $$x, y, z \in X$$ such that $$\partial(x, y) = 2$$, $$\partial(x, z) = i$$, $$\partial(y, z) = i$$ we have $$\alpha_i + \beta_i | \operatorname{\Gamma}_1(x) \cap \operatorname{\Gamma}_1(y) \cap \operatorname{\Gamma}_{i - 1}(z) | = | \operatorname{\Gamma}_{i - 1}(x) \cap \operatorname{\Gamma}_{i - 1}(y) \cap \operatorname{\Gamma}_1(z) |$$; (ii) For all $$x \in X$$ there exist up to isomorphism exactly two irreducible modules for the Terwilliger algebra $$T(x)$$ with endpoint two, and these modules are thin.

##### MSC:
 05E30 Association schemes, strongly regular graphs 05C12 Distance in graphs
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##### References:
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