论文标题
多边形环和扭曲的多边形环的沙珀群
The sandpile group of polygon rings and twisted polygon rings
论文作者
论文摘要
令$ c_ {k_1}，\ ldots，c_ {k_n} $是$ k_i \ geq 2 $ vertices（$ 1 \ le I \ le i \ le n $）。通过以线性顺序将这些$ N $循环连接在一起，我们获得了一个称为多边形链的图。通过将这些$ n $循环连接在一起，我们获得了一个图，如果可以将其嵌入到平面上，该图被称为多边形环。如果可以将其嵌入到Möbius频带上，则称为扭曲的多边形环。众所周知，多边形链的沙珀群始终是循环的。此外，存在边缘发生器。在本文中，我们不仅表明任何（扭曲的）多边形环的沙子基团最多可以由三个边缘产生，而且还可以在这些边缘中给出显式的关系矩阵。因此，我们获得了一种统一的方法来计算任意（扭曲的）多边形环的沙珀组，以及（扭曲）多边形环的跨越树的数量。作为一种应用，我们计算了多边环的几个无限家族的沙门群，其中包括一些以前通过临时方法进行的，例如，概括性的轮子图，梯子和möbius梯子。
Let $C_{k_1}, \ldots, C_{k_n}$ be cycles with $k_i\geq 2$ vertices ($1\le i\le n$). By attaching these $n$ cycles together in a linear order, we obtain a graph called a polygon chain. By attaching these $n$ cycles together in a cyclic order, we obtain a graph, which is called a polygon ring if it can be embedded on the plane; and called a twisted polygon ring if it can be embedded on the Möbius band. It is known that the sandpile group of a polygon chain is always cyclic. Furthermore, there exist edge generators. In this paper, we not only show that the sandpile group of any (twisted) polygon ring can be generated by at most three edges, but also give an explicit relation matrix among these edges. So we obtain a uniform method to compute the sandpile group of arbitrary (twisted) polygon rings, as well as the number of spanning trees of (twisted) polygon rings. As an application, we compute the sandpile groups of several infinite families of polygon rings, including some that have been done before by ad hoc methods, such as, generalized wheel graphs, ladders and Möbius ladders.
