On the extremal Wiener indices of graphs and Wiener (Hosoya) polynomials of capped nanotubes

分子拓扑指数~(topologicalindex)是定义在化合物分子图上的数值不变量~(numericaldescriptor),它与相应化合物的物理、化学和生物性质密切相关,因而被广泛应用于确定“量子结构性质关系（QSPR）”和“量子结构活性关系（QSAR）”,进而用于预测新的化合物的物理、化学和生物性质.拓扑指数在组合优化、网络优化等领域也有广泛应用. Wiener指数最初是由~HaroldWiener在~1947年为了确定烷烃的沸点而引入的一个分子拓扑指数.对于一个连通图~$G$，$G$的~Wiener指数被定义为~$G$中所有顶点对（无序）的距离之和.Wiener指数在通讯、设备定位...Molecular topological indices are the numerical descriptors definedon molecular graphs of compounds, which are closely related tophysical, chemical and biological properties of the correspondingcompounds. So molecular topological indices have been widely appliedto determine the “ quantitative structure-property relations(QSPR)” and “quantitative structure-activity relations (QSAR)”,and to predict ...学位：理学博士院系专业：数学科学学院数学与应用数学系_应用数学学号：1702005140301

Hosoya polynomials of the capped zig-zag nanotubes

The Hosoya polynomial of a graph G is defined as H (G, x) = Sigma(k >= 0) d(G, k)x(k), where d(G, k) is the number of the vertex pairs at distance k in G. The calculation of Hosoya polynomials of molecular graphs is a significant topic because some important molecular topological indices such as Wiener index, hyper-Wiener index, and Wiener vector, can be obtained from Hosoya polynomials. Hosoya polynomials of zig-zag open-ended nanotubes have been given by Xu and Zhang et al. A capped zig-zag nanotube T(p,q)[C, D; a] consists of a zig-zag open-ended nanotube T (p, q) and two caps C and D with the relative position a between C and D. In this paper, we give a general formula for calculating Hosoya polynomial of any capped zig-zag nanotube. By the formula, Hosoya polynomial of any capped zig-zag nanotube can be deduced. Furthermore, it is also shown that any two non-isomorphic capped zig-zag nanotube T(p, q)[C, D; a(1)], T (p, q)[C, D; a(2)] with q >= q* >= p + 1 have the same Hosoya polynomial, where q* is a integer which depends on structures of C and D.NSFC [10831001, 11171279, 11001113]; Natural Science Foundation of Fujian Province of China [2010J01008]; FJCEF [JA11163