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On the Terwilliger algebra of bipartite distance-regular graphs with $$\Delta_{2}=0$$ and $$c_{2}=1$$. (English) Zbl 1331.05237
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 $$x \in X$$ and for $$0 \leq i \leq D$$, let $$\operatorname{\Gamma}_i(x)$$ denote the set of vertices in $$X$$ that are distance $$i$$ from vertex $$x$$. Define a parameter $$\operatorname{\Delta}_2$$ in terms of the intersection numbers by $$\operatorname{\Delta}_2 = (k - 2)(c_3 - 1) -(c_2 - 1) p_{22}^2$$. We first show that $$\operatorname{\Delta}_2 = 0$$ implies that $$D \leq 5$$ or $$c_2 \in \{1, 2 \}$$. 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 $$x$$. By the endpoint of an irreducible $$T$$-module $$W$$ we mean $$\min \{i | E_i^\ast W \neq 0 \}$$.
In this paper we assume $$\Gamma$$ has the property that for $$2 \leq i \leq D - 1$$, there exist complex scalars $$\alpha_i$$, $$\beta_i$$ such that for all $$x, y, z \in X$$ with $$\partial(x, y) = 2$$, $$\partial(x, z) = i$$, $$\partial(y, z) = i$$, we have $$\alpha_i + \beta_i | \operatorname{\Gamma}_1(x) \cap \operatorname{\Gamma}_1(y) \cap \operatorname{\Gamma}_{i - 1}(z) | = | \operatorname{\Gamma}_{i - 1}(x) \cap \operatorname{\Gamma}_{i - 1}(y) \cap \operatorname{\Gamma}_1(z) |$$. We additionally assume that $$\operatorname{\Delta}_2 = 0$$ with $$c_2 = 1$$.
Under the above assumptions we study the algebra $$T$$. We show that if $$\Gamma$$ is not almost 2-homogeneous, then up to isomorphism there exists exactly one irreducible $$T$$-module with endpoint 2. We give an orthogonal basis for this $$T$$-module, and we give the action of $$A$$ on this basis.

##### MSC:
 05E30 Association schemes, strongly regular graphs 05C50 Graphs and linear algebra (matrices, eigenvalues, etc.) 05C12 Distance in graphs
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