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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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