The sandpile group of a thick cycle graph

Diane Christine Alar (Department of Mathematics, University of California, Santa Barbara)
Jonathan Celaya (R&D Operations Research, Penn State University)
Luis David Garcia Puente (Department of Mathematics and Computer Science, Colorado College)
Micah Henson (Department of Applied Mathematics, University of Washington)
Ashley K. Wheeler (School of Mathematics, Georgia Institute of Technology)

Electronic Journal of Graph Theory and Applications (EJGTA)
Vol 10, No 2 (2022): Electronic Journal of Graph Theory and Applications

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.