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The local eigenvalues of a bipartite distance-regular graph. (English) Zbl 1304.05095
Summary: We consider a bipartite distance-regular graph \(\Gamma\) with vertex set \(X\), diameter \(D \geq 4\), and valency \(k \geq 3\). For \(0 \leq i \leq D\), let \(\Gamma_i(x)\) denote the set of vertices in \(X\) that are distance \(i\) from vertex \(x\). We assume there exist scalars \(r, s, t \in \mathbb{R}\), not all zero, such that \[ r | \Gamma_1(x) \cap \Gamma_1(y) \cap \Gamma_2(z) | + s | \Gamma_2(x) \cap \Gamma_2(y) \cap \Gamma_1(z) | + t = 0 \]
for all \(x, y, z \in X\) with path-length distances \(\partial(x, y) = 2\), \(\partial(x, z) = 3\), \(\partial(y, z) = 3\). Fix \(x \in X\), and let \(\Gamma_2^2\) denote the graph with vertex set \(\widetilde{X} = \{y \in X \mid \partial(x, y) = 2 \}\) and edge set \(\widetilde{R} = \{y z \mid y, z \in \widetilde{X}, \partial(y, z) = 2 \}\).
We show that the adjacency matrix of the local graph \(\Gamma_2^2\) has at most four distinct eigenvalues. We are motivated by the fact that our assumption above holds if \(\Gamma\) is \(Q\)-polynomial.

MSC:
05C50 Graphs and linear algebra (matrices, eigenvalues, etc.)
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