论文标题
子树和独立子集中的独立子集和带有固定段序列的独立图和独立图
Subtrees and independent subsets in unicyclic graphs and unicyclic graphs with fixed segment sequence
论文作者
论文摘要
在拓扑指数的研究中，两个负相关是众所周知的：在子树的数量和维也纳指数（距离之和）之间，以及Merrifield-Simmons指数（独立顶点子集的数量）和Hosoya Index（独立边缘子集的数量）之间。也就是说，在某些类别的图中，最大化一个索引的极端图通常最小化另一个索引，反之亦然。在本文中，我们首先研究了带有给定的周围的单周期图和独环图中的子树的数量，通过与已知结果进行比较，进一步证实了其与Wiener指数相反的行为。然后，我们考虑具有给定段序列的独立图，并以最大子树数来表征极端结构。此外，我们表明，相对于Wiener索引，这些图不是极端。我们还确定了具有给定段序列的独立顶部图中独立顶点子集数量的极端结构，并表明它们与独立边缘子集的数量相对于极端。这些结果可能是第一个示例，即这两个索引之间的极端结构中上述负相关失败。
In the study of topological indices two negative correlations are well known: that between the number of subtrees and the Wiener index (sum of distances), and that between the Merrifield-Simmons index (number of independent vertex subsets) and the Hosoya index (number of independent edge subsets). That is, among a certain class of graphs, the extremal graphs that maximize one index usually minimize the other, and vice versa. In this paper, we first study the numbers of subtrees in unicyclic graphs and unicyclic graphs with a given girth, further confirming its opposite behavior to the Wiener index by comparing with known results. We then consider the unicyclic graphs with a given segment sequence and characterize the extremal structure with the maximum number of subtrees. Furthermore, we show that these graphs are not extremal with respect to the Wiener index. We also identify the extremal structures that maximize the number of independent vertex subsets among unicyclic graphs with a given segment sequence, and show that they are not extremal with respect to the number of independent edge subsets. These results may be the first examples where the above negative correlation failed in the extremal structures between these two pairs of indices.
