Sandpile groups for cones over trees
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866929480269824000 |
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| author | Reiner, Victor Smith, Dorian |
| author_facet | Reiner, Victor Smith, Dorian |
| contents | Sandpile groups are a subtle graph isomorphism invariant, in the form of a finite abelian group, whose cardinality is the number of spanning trees in the graph. We study their group structure for graphs obtained by attaching a cone vertex to a tree. For example, it is shown that the number of generators of the sandpile group is at most one less than the number of leaves in the tree. For trees on a fixed number of vertices, the paths and stars are shown to provide extreme behavior, not only for the number of generators, but also for the number of spanning trees, and for Tutte polynomial evaluations that count the recurrent sandpile configurations by their numbers of chips. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_15453 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Sandpile groups for cones over trees Reiner, Victor Smith, Dorian Combinatorics 05C50, 05C25 Sandpile groups are a subtle graph isomorphism invariant, in the form of a finite abelian group, whose cardinality is the number of spanning trees in the graph. We study their group structure for graphs obtained by attaching a cone vertex to a tree. For example, it is shown that the number of generators of the sandpile group is at most one less than the number of leaves in the tree. For trees on a fixed number of vertices, the paths and stars are shown to provide extreme behavior, not only for the number of generators, but also for the number of spanning trees, and for Tutte polynomial evaluations that count the recurrent sandpile configurations by their numbers of chips. |
| title | Sandpile groups for cones over trees |
| topic | Combinatorics 05C50, 05C25 |
| url | https://arxiv.org/abs/2402.15453 |