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An \(A\)-invariant subspace for bipartite distance-regular graphs with exactly two irreducible \(T\)-modules with endpoint 2, both thin. (English) Zbl 1404.05042
Summary: Let \(\Gamma\) denote a bipartite distance-regular graph with vertex set \(X\), diameter \(D \geq 4\), and valency \(k \geq 3\). Let \(\mathbb{C}^X\) denote the vector space over \(\mathbb{C}\) consisting of column vectors with entries in \(\mathbb{C}\) and rows indexed by \(X\). For \(z \in X\), let \(\widehat{z}\) denote the vector in \(\mathbb{C}^X\) with a 1 in the \(z\)-coordinate, and 0 in all other coordinates. Fix a vertex \(x\) of \(\Gamma\) and let \(T = T(x)\) denote the corresponding Terwilliger algebra. Assume that up to isomorphism there exist exactly two irreducible \(T\)-modules with endpoint 2, and they both are thin. Fix \(y \in X\) such that \(\partial(x,y)=2\), where \(\partial\) denotes path-length distance. For \(0 \leq i\), \(j \leq D\) define \(w_{ij}=\sum \widehat{z}\), where the sum is over all \(z \in X\) such that \(\partial (x,z)=i\) and \(\partial (y,z)=j\). We define \(W=\operatorname{span}\{w_{ij} \mid 0 \leq i,j \leq D\}\). In this paper we consider the space \(MW=\operatorname{span}\{mw \mid m \in M, w \in W\}\), where \(M\) is the Bose-Mesner algebra of \(\Gamma \). We observe that \(MW\) is the minimal \(A\)-invariant subspace of \(\mathbb{C}^X\) which contains \(W\), where \(A\) is the adjacency matrix of \(\Gamma\). We show that \(4D-6 \leq \operatorname{dim}(MW) \leq 4D-2\). We display a basis for \(MW\) for each of these five cases, and we give the action of \(A\) on these bases.

MSC:
05C12 Distance in graphs
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