具有最大Clar数的极值富勒烯图
Extremal fullerene graphs with the maximum Clar number
摘要：
一个富勒烯图是一个三次3连通的平面图，它有（正好12个）五边形面和六边形面。设$F_n$是具有$n$顶点的富勒烯图.如果$F_n$有一个完美的匹配，它在$\mathcal H$中的每个六边形上和下交替，则$F_n$的一组互不相交的六边形$\mathcal H$是一个六位体模式。$F_n$的六位模式的最大基数是$F_n$的Clar数。结果表明，Clar数不超过$\lfloor\frac {n-12} 6\rfloor$。许多有实验证据的富勒烯都达到了这个上界，例如$\text{C}_{60}$和$\text{C}_{70}$。本文刻画了Clar数等于$\frac{n-12} 6$的极富勒烯图.通过刻画，我们证明了包含$\text{C}_{60}$在内的60个顶点的富勒烯图中，有18个达到最大Clar数8，并且我们构造了所有这些极值富勒烯图.
英文摘要：
A fullerene graph is a cubic 3-connected plane graph with (exactly 12) pentagonal faces and hexagonal faces. Let $F_n$ be a fullerene graph with $n$ vertices. A set $\mathcal H$ of mutually disjoint hexagons of $F_n$ is a sextet pattern if $F_n$ has a perfect matching which alternates on and off each hexagon in $\mathcal H$. The maximum cardinality of sextet patterns of $F_n$ is the Clar number of $F_n$. It was shown that the Clar number is no more than $\lfloor\frac {n-12} 6\rfloor$. Many fullerenes with experimental evidence attain the upper bound, for instance, $\text{C}_{60}$ and $\text{C}_{70}$. In this paper, we characterize extremal fullerene graphs whose Clar numbers equal $\frac{n-12} 6$. By the characterization, we show that there are precisely 18 fullerene graphs with 60 vertices, including $\text{C}_{60}$, achieving the maximum Clar number 8 and we construct all these extremal fullerene graphs.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
