The sandpile group of polygon rings and twisted polygon rings

TL;DR

This paper provides a uniform method to compute the sandpile group of polygon rings and twisted polygon rings, including explicit generators and relations, with applications to various infinite families like wheels, ladders, and Möbius ladders.

Contribution

It establishes that the sandpile group of any polygon ring or twisted polygon ring can be generated by at most three edges and provides an explicit relation matrix, enabling systematic computation.

Findings

Sandpile groups of polygon rings are generated by at most three edges.

Explicit relation matrices for the generators are derived.

The method applies to infinite families like wheels, ladders, and Möbius ladders.

Abstract

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 M\"{o}bius 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 M\"{o}bius ladders.