| Abstract | 六角系统是一个没有割点的有限连通平面图, 其每个内面边界都是单位边长的正六角形. 六角系统的一个几何凯库勒结构 (GKS) 相当于图的完美匹配, 可对应一个代数凯库勒结构 (AKS), 它是定义在六角形上的函数: 若这个几何凯库勒结构中的一个双键同时被两个六角形共用, 则它对这两个六角形分别贡献 1; 若一个双键仅属于一个六角形, 则它对这个六角形贡献 2. 这样每个六角形上的函数值就是它六条边中双键对它的贡献和, 我们称为 Randi´c 数或 π-电子数. AKS 最初是由 Randi´c 等引入的, Gutman 等证明了除了单个六角形外的 cata-型六角系统(不存在内点, 即三个六角形共用一个顶点)的 AKS 与 GKS 是一一对应的, 但是这样的结果并不总是成立的, T. Balaban 等也考虑了冠状系统的 π- 电子在六角形上的分配.
冠状系统是六角系统的一个连通子图, 其每条边都位于一个六角形上, 它至少包含一个非六角形的内面(我们称之为洞). 如果它不包含内点, 则称为 cata-型冠状系统. 一个 cata-型冠状系统称为无分叉的(也称为 primitive 冠状系统), 如果它恰好有两个相邻六角形. 否则称为有分叉的.本文主要考虑了单洞 cata-型冠状系统的代数和几何凯库勒结构计数之间的关系. 对 primitive 冠状系统, 我们表明了若它由 L2和 A2模式的六角形交替连接和全是由 A2模式的六角形连接而成其 AKS 的个数比 GKS 的个数恰少 2, 其它情形恰少 1. 对有分叉的单洞 cata-型冠状系统, 得到了: (1) 由 L2和 A2模式的六角形交替连接形成一个环并加上一些分叉得到的冠状系统, 它的 AKS 的个数比GKS 的个数恰少每个分叉上的完美匹配个数的乘积; (2) 由全是 A2模式的六角形连接形成一个环并加上一些分叉得到的冠状系统: 若分叉属于同类, 结果与 (1)相同, 若分叉属于不同类, 它们之间的 AKS 和 GKS 是一一对应的. (3) 除了以上两类外, 它们之间的 AKS 和 GKS 是一一对应的. |
| Other Abstract | A hexagonal system is a connected plane graph without cut vertices in which each interior face is a regular hexagon of side length one. A geometric Kekulé Structure (GKS) of a hexagonal system is equivalent to a perfect matching, which
corresponds to an algebraic Kekulé Structure (AKS). In fact, it is a function that assigns to each hexagon of a hexagonal system one integer: each double bond in GKS that belongs to only one hexagon contributes 2 to that hexagon and each double bond that is shared by two hexagons contributes 1 to each one of these two hexagons. So the function value on each hexagon is the sum of the contribution of the double bond in it. Let us call it Randi´c number or π-electron number. Gutman et al. proved that there exists an one-to-one correspondence between GKS and AKS of any cata-condensed hexagon system(no three hexagon share a common vertex) with at least 2 hexagons. But the result does not always hold. T. Balaban et al. also considered the distribution of π-electrons on hexagons of coronoid systems.
A coronoid system is a connected subgraph G of a hexagonal so that each edge belongs to a hexagon of G and G contains at least one non-hexagonal inner face (we called a hole). We call it cata-condensed coronoid if no three hexagon
share a common vertex. A cata-condensed coronoid is unbranched(or primitive coronoid ), if each hexagon is adjacent to exactly two hexagons. Otherwise it is
branched.
This paper mainly considers the relationship between the count of the GKS of a single hole condensed coronoid. We demonstrated that if the primitive coronoid is connected by L2 and A2 mode alternately or connected by all A2 modes, then the number of AKS is equal to the number of GKS minus 2. The number of AKS is equal to the number of GKS minus 1 for other primitive coronoid. We obtained the following results for branched cata-condensed coronoid: (1) the number of AKS is equal to the number of GKS minus the product of each perfect matching of all branches if it connected by L2 mode, A2 mode alternately and added some branches; (2) the branched cata-condensed coronoid that connected by all A2modes and added some branches, the results is same as (1) if branches belong to same class, there exists an one-to-one correspondence between AKS and GKS if branches belong to different class; (3) there exists an one-to-one correspondence between AKS and GKS for other branched cata-condensed coronoid. |
