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The local structure of a bipartite distance-regular graph. (English) Zbl 0940.05074

Summary: We consider a bipartite distance-regular graph \(\Gamma= (X,E)\) with diameter \(d\geq 3\). We investigate the local structure of \(\Gamma\), focusing on those vertices with distance at most 2 from a given vertex \(x\). To do this, we consider a subalgebra \({\mathcal R}={\mathcal R}(x)\) of \(\text{Mat}_{\widetilde X}(\mathbb{C})\), where \(\widetilde X\) denotes the set of vertices in \(X\) at distance \(2\) from \(x\). \({\mathcal R}\) is generated by matrices \(\widetilde A\), \(\widetilde J\), and \(\widetilde D\) defined as follows. For all \(y,z\in\widetilde X\), the \((y,z)\)-entry of \(\widetilde A\) is \(1\) if \(y\), \(z\) are at distance \(2\), and \(0\) otherwise. The \((y,z)\)-entry of \(\widetilde J\) equals \(1\), and the \((y,z)\)-entry of \(\widetilde D\) equals the number of vertices of \(X\) adjacent to each of \(x\), \(y\), and \(z\). We show that \({\mathcal R}\) is commutative and semisimple, with dimension at least \(2\). We assume that \(\dim{\mathcal R}\) is one of \(2\), \(3\), or \(4\), and explore the combinatorial implications of this. We are motivated by the fact that if \(\Gamma\) has a \(Q\)-polynomial structure, then \(\dim{\mathcal R}\leq 4\).

MSC:

05E30 Association schemes, strongly regular graphs
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