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On bipartite distance-regular graphs with exactly one non-thin $$T$$-module with endpoint two. (English) Zbl 1365.05078
Summary: Let $$\varGamma$$ denote a bipartite distance-regular graph with diameter $$D \geq 4$$ and valency $$k \geq 3$$. Let $$X$$ denote the vertex set of $$\varGamma$$, and let $$A$$ denote the adjacency matrix of $$\varGamma$$. For $$x \in X$$ and for $$0 \leq i \leq D$$, let $$\varGamma_i(x)$$ denote the set of vertices in $$X$$ that are distance $$i$$ from vertex $$x$$. Define a parameter $$\varDelta_2$$ in terms of the intersection numbers by $$\varDelta_2 = (k - 2)(c_3 - 1) -(c_2 - 1) p_{22}^2$$.
For $$x \in X$$ let $$T = T(x)$$ denote the subalgebra of $$\mathrm{Mat}_X(\mathbb{C})$$ generated by $$A$$, $$E_0^\ast$$, $$E_1^\ast$$, …, $$E_D^\ast$$, where for $$0 \leq i \leq D$$, $$E_i^\ast$$ represents the projection onto the $$i$$th subconstituent of $$\varGamma$$ with respect to $$x$$. We refer to $$T$$ as the Terwilliger algebra of $$\varGamma$$ 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 an irreducible $$T$$-module $$W$$ we mean $$\min\{i \mid E_i^\ast W \neq 0 \}$$.
Fix $$x \in X$$ and assume that $$\varGamma$$ has, up to isomorphism, exactly one irreducible $$T$$-module $$W$$ with endpoint 2, and that $$W$$ is non-thin with $$\dim(E_2^\ast W) = 1$$, $$\dim(E_{D - 1}^\ast W) \leq 1$$ and $$\dim(E_i^\ast W) \leq 2$$ for $$3 \leq i \leq D$$. We prove that for $$2 \leq i \leq D$$, there exist complex scalars $$\alpha_i$$, $$\beta_i$$ such that $$| \varGamma_{i - 1}(x) \cap \varGamma_{i - 1}(y) \cap \varGamma_1(z) | = \alpha_i + \beta_i | \varGamma_1(x) \cap \varGamma_1(y) \cap \varGamma_{i - 1}(z) |$$ for all $$y \in \varGamma_2(x)$$ and $$z \in \varGamma_i(x) \cap \varGamma_i(y)$$. Furthermore, we prove $$\varDelta_2 = 0$$ and either $$D = 5$$ or $$c_2 \in \{1, 2 \}$$. We show there exist integers $$3 \leq f \leq \ell \leq D - 2$$ such that $$\dim(E_i^\ast W) = 2$$ if and only if $$f \leq i \leq \ell$$.

MSC:
 05C12 Distance in graphs
Keywords:
Terwilliger algebra
Full Text:
References:
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