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arXiv · 1912.06919

Sandpile Groups of Cayley Graphs of $\mathbb{F}_2^r$

Jiyang Gao·Jared Marx-Kuo·Vaughan McDonald·Chi Ho Yuen
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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 $\mathbb{F}_2^r$, 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.

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Jiyang Gao, Jared Marx-Kuo, Vaughan McDonald, Chi Ho Yuen. 2024-04-17. Sandpile Groups of Cayley Graphs of $\mathbb{F}_2^r$. https://arxiv.org/abs/1912.06919

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