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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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