随机正十边形链的Hosoya指数研究

On the Hosoya Index of Random Regular Decagonal Chains

摘要: 化学图论是化学与图论交叉的一个活跃领域。拓扑指数是不随图顶点编序而改变的图不变量，能将复杂分子结构转化为可计算的数值，是化学图论中建立分子结构与性质定量关系的核心工具。本文研究的Hosoya指数是化学图论中极具价值的一类拓扑指数，由日本化学家Hosoya于1971年提出，定义为图G中所有匹配的数目之和。其数值变化与烷烃的沸点、分子稳定性等物理化学性质密切相关。本文研究随机正十边形链关于Hosoya指数的期望。在该链中，两个相邻的十边形通过割边以随机方式连接。通过概率图方法将随机结构转化为可量化的数学结构，并建立递推公式，推导出随机正十边形链中Hosoya指数的期望表达式。

Abstract: Chemical graph theory is an active interdisciplinary field bridging chemistry and graph theory. Topological indices are graph invariants that do not change with vertex labeling; they convert complex molecular structures into computable numerical values and serve as core tools for establishing quantitative structure-property relationships in chemical graph theory. The Hosoya index studied in this paper is a highly valuable type of topological index in chemical graph theory. Proposed by Japanese chemist Hosoya in 1971, it is defined as the total number of matchings in a graph G. Its numerical variation is closely related to physicochemical properties of alkanes, such as boiling point and molecular stability. This paper investigates the expectation of the Hosoya index for random regular decagonal chains. In such a chain, two adjacent decagons are connected by a cut edge in a random manner. By employing probabilistic graph methods, the random structure is transformed into a quantifiable mathematical structure. A recurrence relation is then established, and the expected expression of the Hosoya index for random regular decagonal chains is derived.