Sandpile Groups of Cayley Graphs of $\mathbb{F}_2^r$
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arXiv
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| Auteurs principaux: | , , , |
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| Format: | Preprint |
| Publié: |
2019
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| _version_ | 1866917642052304896 |
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| author | Gao, Jiyang Marx-Kuo, Jared McDonald, Vaughan Yuen, Chi Ho |
| author_facet | Gao, Jiyang Marx-Kuo, Jared McDonald, Vaughan Yuen, Chi Ho |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1912_06919 |
| institution | arXiv |
| publishDate | 2019 |
| record_format | arxiv |
| spellingShingle | Sandpile Groups of Cayley Graphs of $\mathbb{F}_2^r$ Gao, Jiyang Marx-Kuo, Jared McDonald, Vaughan Yuen, Chi Ho Combinatorics Rings and Algebras 05C50 (Primary) 15B36 (Secondary) 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. |
| title | Sandpile Groups of Cayley Graphs of $\mathbb{F}_2^r$ |
| topic | Combinatorics Rings and Algebras 05C50 (Primary) 15B36 (Secondary) |
| url | https://arxiv.org/abs/1912.06919 |