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

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.
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Taxonomy
TopicsAdvanced Graph Theory Research · Graph Labeling and Dimension Problems · Interconnection Networks and Systems
