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 \(f \leq i \leq \ell\).


05C12 Distance in graphs
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