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2-homogeneous bipartite distance-regular graphs. (English) Zbl 0958.05143
Summary: Let $$\Gamma$$ denote a bipartite distance-regular graph with diameter $$D\geq 3$$ and valency $$k\geq 3$$. $$\Gamma$$ is said to be 2-homogeneous whenever for all integers $$i$$ $$(1\leq i\leq D-1)$$ and for all vertices $$x$$, $$y$$, $$z$$ at distance $$\partial(x, y)= 2$$, $$\partial(x,z)= i$$, $$\partial(y,z)= i$$, the number $$\gamma_i$$ of vertices adjacent to both $$x$$ and $$y$$ and at distance $$i-1$$ from $$z$$ is a constant depending only upon $$i$$.
We characterize the 2-homogeneous property in three ways. These characterizations involve the intersection numbers, the eigenvalues, and the Krein parameters, respectively.
First, we obtain a sequence of inequalities involving the intersection numbers of $$\Gamma$$. We show that equality is attained in every case if and only if $$\Gamma$$ is 2-homogeneous. Second, we obtain a number of inequalities involving the eigenvalues of $$\Gamma$$. We show that equality is attained in any one of them if and only if equality is attained in all of them if and only if $$\Gamma$$ is 2-homogeneous. Third, we show that the following are equivalent: (i) $$\Gamma$$ is 2-homogeneous; (ii) $$\Gamma$$ is an antipodal 2-cover and Q-polynomial; and (iii) $$\Gamma$$ has a Q-polynomial structure for which the Krein parameters $$q^i_{1i}= 0$$ $$(0\leq i\leq D)$$.

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