六角链及环状六角链的 Clar 覆盖多项式的单峰性 Alternative Title The unimodality of the Clar covering polynomials of the hexagonal chains and cyclic hexagonal chains 崔杨 Thesis Advisor 张和平 2013-05-28 Degree Grantor 兰州大学 Place of Conferral 兰州 Degree Name 硕士 Keyword 六角链 环状六角链 Clar 覆盖多项式 对数凹 单峰性 Abstract 六角系统 $H$ 的 $Clar$ 覆盖多项式 $P(H, \omega)$ 是1993年由张和平和张福基首先提出的, 它很好的统一了六角系统一些重要的拓扑指标, 如 $Kekule$ 结构的个数 $\sigma(H, 0)$, 第一类 $Herdon$ 数 $\sigma(H, 1)$ 以及 $Clar$ 数 $C(H)$ 等. 他们还于1996年提出了一个猜想: $P(H, \omega)$ 的系数是单峰的, 同时证明了当六角系统满足 $1\leqslant C(H)\leqslant5$ 时, 猜想正确. 本文利用六角链的 $Clar$ 覆盖多项式与其 $Gutman$ 树的 匹配多项式之间的关系及对数凹的方法证明了猜想对六角链是成立的. 同时, 对此方法进行了推广, 证明了偶数个片段的环状六角链的 $Clar$ 覆盖多项式是单峰的, 偶数个片段的环状六角链是原始冠的推广, 从而得出原始冠的相关结果. 最后, 本文根据奇数个片段的锯齿型环状六角链的 $Clar$ 覆盖多项式的显式表达式证明了其系数的单峰性. Other Abstract $P(H, \omega)$, the Clar covering polynomial of the hexagonal system H, is firstly proposed by Heping Zhang and Fuji Zhang in 1993 which can unify some important topological indices of the hexagonal systems, such as the number of $Kekule$ structures $\sigma(H, 0)$, first Herdon number $\sigma(H, 1)$, Clar number $C(H)$, etc. They also posed the following conjecture in 1996: The coefficients of $P(H, \omega)$ are unimodal. They showed that the conjecture is true for any hexagonal systems H with $1\leqslant C(H)\leqslant 5$ at the same time. In this paper, we show the conjectrue is true for hexagonal chains by the relationship between the Clar covering polynomial of a hexagonal chain and matching polynomial of its Gutman tree and the means of log-concave. Then by generalizing this method, we obtain that the Clar covering polynomial of any cyclic hexagonal chain with even segments which is the spreading of the primitive coronoid is unimodal, thus we get some relevant results about primitive coronoids. At last we prove that the Clar covering polynomial of a zigzag cyclic hexagonal chain with odd segments is unimodal by its explicit expression. URL 查看原文 Language 中文 Document Type 学位论文 Identifier https://ir.lzu.edu.cn/handle/262010/224865 Collection 数学与统计学院 Recommended CitationGB/T 7714 崔杨. 六角链及环状六角链的 Clar 覆盖多项式的单峰性[D]. 兰州. 兰州大学,2013.
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