The Sandpile Group of a Cone Over a Bi-Coconut Tree

TL;DR

This paper computes the sandpile group and spanning tree number for the cone over a bi-coconut tree, extending previous work and providing examples of trees with cyclic sandpile groups and unbounded leaves.

Contribution

It generalizes the computation of sandpile groups to bi-coconut trees and answers a question about trees with cyclic sandpile groups and increasing leaves.

Findings

Computed the spanning tree number for the cone over a bi-coconut tree.

Determined the structure of the sandpile group for these graphs.

Provided examples of trees with cyclic sandpile groups and unbounded leaves.

Abstract

The sandpile group of a connected graph is a finite abelian group whose cardinality is the number of spanning trees in the graph. We compute the spanning tree number and sandpile group structure for the cone over a bi-coconut tree, generalizing work of Reiner and Smith on the cone over a coconut tree. We also answer one of their questions, by exhibiting a family of trees whose sandpile groups are all cyclic but their number of leaves grows without bound.