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

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 $G$, 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.