Sandpile Groups of Cayley Graphs of $\mathbb{F}_2^r$
Jiyang Gao, Jared Marx-Kuo, Vaughan McDonald, Chi Ho Yuen

TL;DR
This paper investigates the structure of sandpile groups of Cayley graphs of _2^r, especially their Sylow-2 components, providing formulas, bounds, and insights into their algebraic and combinatorial properties.
Contribution
It extends previous work by analyzing the Sylow-2 parts of sandpile groups for these graphs, offering new formulas and bounds, particularly for hypercubes.
Findings
Number of Sylow-2 cyclic factors for generic Cayley graphs
Bounds for non-generic Cayley graphs' Sylow-2 components
Exact formulas for Sylow-2 factors in hypercubes
Abstract
The sandpile group of a connected graph , 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 , focusing on their poorly understood Sylow- component. We find the number of Sylow- 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- cyclic factors. In the case of hypercubes, we give exact formulae for the largest Sylow- cyclic factors. Some key ingredients of our work include the natural ring structure on these sandpile groups from representation theory, and calculation of the -adic valuations of binomial sums via the…
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Taxonomy
TopicsGeometric and Algebraic Topology · Topological and Geometric Data Analysis · Graph theory and applications
