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arXiv · 2505.05392

Generalized chip firing and critical groups of arithmetical structures on trees

Kassie Archer·Alexander Diaz-Lopez·Darren Glass·Joel Louwsma
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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.

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Kassie Archer, Alexander Diaz-Lopez, Darren Glass, Joel Louwsma. 2025-07-03. Generalized chip firing and critical groups of arithmetical structures on trees. https://arxiv.org/abs/2505.05392

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