## 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:

 5e+30 Association schemes, strongly regular graphs

### Keywords:

bipartite distance-regular graph; local structure; distance
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### References:

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