冠状系统和纳米管的共振图
The resonance graphs of coronoid systems and nanotubes
AI总结：
本文提出冠状系统完美匹配属于共振图同一连通分支的图论判据，构造共振图连通的纳米管反驳了Tratnik等人的相关猜想。
AI中文摘要：
六边形系统的共振图是连通的，这表明可以通过沿六边形的一系列翻转将一个完美匹配转换为任意其他完美匹配。然而，带有孔的冠状系统的共振图不一定连通。Saldanha等人（《离散计算几何》14卷，1995年，第207-233页）使用同调和上同调理论，为平面上四格区域的两种 tiling（铺砌）属于翻转图同一连通分支获得了三种版本的判据。受组合版本的启发，本文采用纯图论方法，以简单不变量——跨孔/外表面割的流——为依据，给出了冠状系统G的两个完美匹配属于其共振图同一连通分支的判据。作为推论，得到了冠状系统共振图连通的判据。本文还讨论了此类判据是否适用于纳米管，并构造了一个共振图连通的纳米管，反驳了Tratnik等人（《MATCH 数学计算化学通讯》74卷，2015年，第175-186页）提出的猜想。
英文摘要：
The resonance graph of a hexagonal system is connected, which shows that a perfect matching can be transformed into any other perfect matchings by a series of flips along hexagons. However, the resonance graph of a coronoid system (with holes) is not necessarily connected. Saldanha et al. (Discrete Comput. Geom. 14 (1995) 207-233) used homology and cohomology theory to obtain three versions of criteria for two tilings of a quadriculated region in the plane to be in the same connected component of the flip graph. Inspiblack by the combinatorial version, in this paper we use a purely graph-theoretical approach to give a criterion in terms of simple invariant\textcolor{black}{---flow} across cuts between holes/exterior face for two perfect matchings of a coronoid system $G$ to be in the same connected component of its resonance graph. As a corollary we obtain a criterion for the resonance graph of a coronoid system to be connected. We also discuss whether such \textcolor{black}{criteria} are applicable to nanotubes, and construct a nanotube whose resonance graph is connected, which disproves a conjecture proposed by Tratnik et al. (MATCH Commun. Math. Comput. Chem. 74 (2015) 175-186).
