The Wiener polarity index of a graph , usually denoted by , is defined as the number of unordered pairs of those vertices of that are at distance 3. A vertex of a tree with degree at least 3 is called a branching vertex. A segment of a tree is a nontrivial path whose end-vertices have degrees different from 2 in and every other vertex (if exists) of has degree 2 in . In this note, the best possible sharp lower bounds on the Wiener polarity index are derived for the trees of fixed order and with a given number of branching vertices or segments, and all the trees attaining this lower bound are characterized.

中文翻译：

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关于具有给定数量的顶点和线段或分支顶点的树木的最小Wiener极性指数的注释

的曲线图的维纳极性指数，通常由表示，被定义为无序对那些顶点的数量是在一棵树的具有至少3度的距离3.顶点被称为分支顶点。树的分段是一条非平凡的路径，其端点的度数不同于2 in，并且其他每个顶点（如果存在）的度数都为2 in 。在本说明中，针对固定顺序并具有给定数量的分支顶点或分段的树，得出了维纳极性指数上可能的最佳锐利下界，并对达到该下界的所有树进行了特征化。