2-Homogeneous bipartite distance-regular graphs
2-Homogeneous bipartite distance-regular graphs
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2-齐次二分距离-正则图
DOI：
10.1016/s0012-365x(97)00226-4
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发表时间：
1998
期刊：
影响因子：
--
通讯作者：
B. Curtin
中科院分区：
文献类型：
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作者：
B. Curtin
设Γ表示一个直径D大于或等于3，价k大于或等于3的二部距离规则图。对于所有整数i（1≤i≤D - 1）和所有顶点x，y， Γ都是2齐次的。在距离∂(x, y) = 2,∂(x, z) = i,∂(y, z) = i处，与x和y相邻且距离z为i - 1处的顶点数γ是一个常数，仅取决于i。在本文中，我们用三种方式表征了2齐次性质。这些特征分别涉及交集数、特征值和Krein参数。首先，我们得到了一个包含Γ相交数的不等式序列。我们证明当且仅当Γ是2齐次的，在任何情况下都是相等的。其次，我们得到了一些涉及Γ的特征值的不等式。我们证明当且仅当Γ是2齐次的，当且仅当所有的相等时，它们中的任何一个都是相等的。第三，我们证明了以下是等价的：(i) Γ是2齐次的；（ii） Γ是对映2-盖和q -多项式；（iii） Γ具有k -多项式结构，其Krein参数q1ii= 0（0≤i≤D）。
Let Γ denote a bipartite distance-regular graph with diameter D ⩾ 3 and valency k ⩾ 3. Γ is said to be 2-homogeneous whenever for all integers i(1⩽i⩽D − 1) and for all vertices x,y.z at distance ∂(x, y) = 2, ∂(x, z) = i, ∂(y, z) = i, the number γ of vertices adjacent to both x and y and at distance i − 1 from z is a constant depending only upon i. In this paper 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 Γ. We show that equality is attained in every case if and only if Γ is 2-homogeneous. Second, we obtain a number of inequalities involving the eigenvalues of Γ. 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 Γ is 2-homogeneous. Third, we show that the following are equivalent: (i) Γ is 2-homogeneous; (ii) Γ is an antipodal 2-cover and Q-polynomial; and (iii) Γ has a Q-polynomial structure for which the Krein parameters q1ii= 0 (0⩽i⩽D).