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Almost 2-homogeneous bipartite distance-regular graphs. (English) Zbl 1002.05069
Summary: Let $$\Gamma= (X,R)$$ denote a bipartite distance-regular graph with diameter $$d\geq 4$$, and fix a vertex $$x$$ of $$\Gamma$$. The Terwilliger algebra of $$\Gamma$$ with respect to $$x$$ is the subalgebra $$T$$ of $$\text{Mat}_X(\mathbb{C})$$ generated by $$A$$, $$E^*_0, E^*_1,\dots, E^*_d$$, where $$A$$ is the adjacency matrix of $$\Gamma$$, and where $$E^*_i$$ denotes the projection onto the $$i$$th subconstituent of $$\Gamma$$ with respect to $$x$$. Let $$W$$ denote an irreducible $$T$$-module. $$W$$ is said to be thin whenever $$\dim E^*_i W\leq 1$$ $$(0\leq i\leq d)$$. The endpoint of $$W$$ is $$\min\{i\mid E^*_i W\neq 0\}$$. It is known that a thin irreducible $$T$$-module of endpoint 2 has dimension $$d-3$$, $$d-2$$, or $$d-1$$.
$$\Gamma$$ is said to be $$2$$-homogeneous whenever for all $$i$$ $$(1\leq i\leq d-1)$$ and for all $$x,y,z\in X$$ with $$\partial(x, y)= 2$$, $$\partial(x, z)= i$$, $$\partial(y, z)= i$$, the number $$|\Gamma_1(x)\cap \Gamma_1(y)\cap \Gamma_{i- 1}(z)|$$ is independent of $$x$$, $$y$$, $$z$$. Nomura has classified the $$2$$-homogeneous bipartite distance-regular graphs. In this paper we study a slightly weaker condition. $$\Gamma$$ is said to be almost $$2$$-homogeneous whenever for all $$i$$ $$(1\leq i\leq d-2)$$ and for all $$x,y,z\in X$$ with $$\partial(x, y)= 2$$, $$\partial(x, z)= i$$, $$\partial(y, z)= i$$, the number $$|\Gamma_1(x)\cap \Gamma_1(y)\cap \Gamma_{i- 1}(z)|$$ is independent of $$x$$, $$y$$, $$z$$. We prove that the following are equivalent: (i) $$\Gamma$$ is almost $$2$$-homogeneous; (ii) $$\Gamma$$ has, up to isomorphism, a unique irreducible $$T$$-module of endpoint $$2$$ and this module is thin. Moreover, $$\Gamma$$ is $$2$$-homogeneous if and only if (i) and (ii) hold and the unique irreducible $$T$$-module of endpoint $$2$$ has dimension $$d- 3$$.

##### MSC:
 05E30 Association schemes, strongly regular graphs 05C12 Distance in graphs
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