## 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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### References:

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