The Sandpile Group of a Thick Cycle Graph

TL;DR

This paper derives an explicit formula for the sandpile group of thick cycle graphs, a class of non-regular multi-graphs, revealing their structure in terms of edge multiplicities and minors of the Laplacian matrix.

Contribution

It provides the first explicit formula for the sandpile groups of thick cycle graphs, extending known results beyond regular and simple graphs.

Findings

Sandpile group of thick cycles is a direct sum of cyclic groups.

GCDs of minors of Laplacian are expressed via monomials in edge multiplicities.

Explicit formula links graph structure to algebraic properties.

Abstract

The majority of graphs whose sandpile groups are known are either regular or simple. We give an explicit formula for a family of non-regular multi-graphs called thick cycles. A thick cycle graph is a cycle where multi-edges are permitted. Its sandpile group is the direct sum of cyclic groups of orders given by quotients of greatest common divisors of minors of its Laplacian matrix. We show these greatest common divisors can be expressed in terms of monomials in the graph's edge multiplicities.