2-homogeneous bipartite distance-regular graphs.

*(English)*Zbl 0958.05143Summary: 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 \(q^i_{1i}= 0\) \((0\leq i\leq D)\).

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 \(q^i_{1i}= 0\) \((0\leq i\leq D)\).

##### Keywords:

bipartite distance-regular graph; characterizations; intersection numbers; eigenvalues; Krein parameters
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##### References:

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