Sandpile groups of Cayley graphs of Fr2

Jiyang Gao, Jared Marx-Kuo, Vaughan McDonald, Chi Ho Yuen*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

Abstract

The sandpile group of a connected graph G, defined to be the torsion part of the cokernel of the graph Laplacian, is a subtle graph invariant with combinatorial, algebraic, and geometric descriptions. Extending and improving previous works on the sandpile group of hypercubes, we study the sandpile groups of the Cayley graphs of (Formula presented.), focusing on their poorly understood Sylow-2 component. We find the number of Sylow-2 cyclic factors for “generic” Cayley graphs and deduce a bound for the non-generic ones. Moreover, we provide a sharp upper bound for their largest Sylow-2 cyclic factors. In the case of hypercubes, we give exact formulae for the largest n–1 Sylow-2 cyclic factors. Some key ingredients of our work include the natural ring structure on these sandpile groups from representation theory, and calculation of the 2-adic valuations of binomial sums via the combinatorics of carries.

Original languageEnglish
Pages (from-to)4459-4479
Number of pages21
JournalCommunications in Algebra
Volume52
Issue number10
DOIs
StatePublished - 2024

Keywords

  • Cayley graph
  • sandpile group

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