The Sandpile Group of Polygon Rings and Twisted Polygon Rings

Let $$C_{k_1}, \ldots , C_{k_n}$$ be cycles with $$k_i\ge 2$$ vertices ( $$1\le i\le n$$ ). By attaching these n together in a linear order, we obtain graph called polygon chain. cyclic graph, which is ring if it can embedded on the plane; and twisted Möbius band. It known that sandpile group of chain always cyclic. Furthermore, there exist edge generators. In this paper, not only show any (twisted) generated by at most three edges, but also give an explicit relation matrix among edges. So uniform method to compute arbitrary rings, as well number spanning trees rings. As application, groups several infinite families including some have been done before ad hoc methods, such as, generalized wheel graphs, ladders ladders.