论文标题
维纳指数，子树的数量和树怪异序列
Wiener index, number of subtrees, and tree eccentric sequence
论文作者
论文摘要
在连接的图$ g $中，顶点$ u $的偏心率是$ u $和距其最远的顶点之间的距离； $ g $的偏心序列是$ g $的偏心率的非续顺序。在本文中，我们确定了将Wiener索引最小化的独特树，即所有无序顶点对之间的距离之和，在具有给定的偏心序列的所有树中。我们表明，同一树以给定的偏心序列最大化了所有树中的子树的数量，从而提供了另一个子树与树木索引数量之间负相关的另一个例子。此外，我们根据偏心序列提供了这两个不变的相应极值的公式。作为我们结果的推论，我们确定了将边缘Wiener索引，顶点边缘索引，Schulz索引（或程度距离）以及具有给定偏心序列的所有树中的Gutman索引最小化的独特树。
The eccentricity of a vertex $u$ in a connected graph $G$ is the distance between $u$ and a vertex farthest from it; the eccentric sequence of $G$ is the nondecreasing sequence of the eccentricities of $G$. In this paper, we determine the unique tree that minimises the Wiener index, i.e. the sum of distances between all unordered vertex pairs, among all trees with a given eccentric sequence. We show that the same tree maximises the number of subtrees among all trees with a given eccentric sequence, thus providing another example of negative correlation between the number of subtrees and the Wiener index of trees. Furthermore, we provide formulas for the corresponding extreme values of these two invariants in terms of the eccentric sequence. As a corollary to our results, we determine the unique tree that minimises the edge Wiener index, the vertex-edge Wiener index, the Schulz index (or degree distance), and the Gutman index among all trees with a given eccentric sequence.
