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.


05E30 Association schemes, strongly regular graphs
05C12 Distance in graphs
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