螺环和多苯基杂环链的Wiener指数
Wiener Indices of Spiro and Polyphenyl Hexagonal Chains
摘要：
连通图$G$的Wiener指数W（G）是G$中所有顶点对之间的距离之和。本文首先给出了一类无支链多螺环分子和多环芳烃的图--螺环和多苯基六方链的Wiener指数的递推或显式计算公式，然后建立了螺环六方链和多苯基六方链的Wiener指数之间的关系，确定了具有n个六边形的螺环六角链和聚苯六角链的Wiener指数极值，并刻画了它们的Wiener指数极值图.一个有趣的结果表明，所有六角链的Wiener指数的平均值正好是三个特殊六角链的Wiener指数的平均值，并且是元链的Wiener指数.
英文摘要：
The Wiener index W(G) of a connected graph $G$ is the sum of distances between all pairs of vertices in G$. In this paper, we first give the recurrences or explicit formulae for computing the Wiener indices of spiro and polyphenyl hexagonal chains, which are graphs of a class of unbranched multispiro molecules and polycyclic aromatic hydrocarbons, then we establish a relation between the Wiener indices of a spiro hexagonal chain and its corresponding polyphenyl hexagonal chain, and determine the extremal values and characterize the extremal graphs with respect to the Wiener index among all spiro and polyphenyl hexagonal chains with n hexagons, respectively. An interesting result shows that the average value of the Wiener indices with respect to the set of all such hexagonal chains is exactly the average value of the Wiener indices of three special hexagonal chains, and is just the Wiener index of the meta-chain.
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