Generalized chip firing and critical groups of arithmetical structures on trees

TL;DR

This paper extends chip firing theory to arithmetical structures on trees, providing bounds, classifications, and constructions for their critical groups, including realizing all finite abelian groups as such critical groups.

Contribution

It introduces a generalized chip firing approach to analyze critical groups on trees, classifies trees with cyclic critical groups, and constructs structures with prescribed critical groups.

Findings

Bound the number of invariant factors of critical groups.

Critical groups are additive under wedge sums of graphs.

Every finite abelian group can be realized as a critical group on some tree.

Abstract

Chip firing provides a way to study the sandpile group (also known as the Jacobian) of a graph. We use a generalized version of chip firing to bound the number of invariant factors of the critical group of an arithmetical structure on a graph. We also show that, under suitable hypotheses, critical groups are additive under wedge sums of graphs with arithmetical structures. These results allow us to relate the number of invariant factors of critical groups associated to any given tree to decompositions of the tree into simpler trees. We use this to classify those trees for which every arithmetical structure has cyclic critical group. Finally, we show how to construct arithmetical structures on trees with prescribed critical groups. In particular, every finite abelian group is realized as the critical group of some arithmetical structure on a tree.