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| Main Authors: | , , , , , |
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| Format: | Preprint |
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2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2406.11183 |
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| _version_ | 1866914836607139840 |
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| author | Diaz-Lopez, Alexander Ha, Brian Harris, Pamela E. Rogers, Jonathan Koss, Theo Smith, Dorian |
| author_facet | Diaz-Lopez, Alexander Ha, Brian Harris, Pamela E. Rogers, Jonathan Koss, Theo Smith, Dorian |
| contents | If G is a finite connected graph, then an arithmetical structure on $G$ is a pair of vectors $(\mathbf{d}, \mathbf{r})$ with positive integer entries such that $(\diag(\mathbf{d}) - A)\cdot \mathbf{r} = \mathbf{0}$, where $A$ is the adjacency matrix of $G$ and the entries of $\mathbf{r}$ have no common factor other than $1$. In this paper, we generalize the result of Archer, Bishop, Diaz-Lopez, García Puente, Glass, and Louwsma on enumerating arithmetical structures on bidents (also called coconut tree graphs $\CT{p}{2}$) to all coconut tree graphs $\CT{p}{s}$ which consists of a path on $p>0$ vertices to which we append $s>0$ leaves to the right most vertex on the path. We also give a characterization of smooth arithmetical structures on coconut trees when given number assignments to the leaf nodes. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_11183 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Arithmetical Structures on Coconut Trees Diaz-Lopez, Alexander Ha, Brian Harris, Pamela E. Rogers, Jonathan Koss, Theo Smith, Dorian Combinatorics 05C50, 05C30 If G is a finite connected graph, then an arithmetical structure on $G$ is a pair of vectors $(\mathbf{d}, \mathbf{r})$ with positive integer entries such that $(\diag(\mathbf{d}) - A)\cdot \mathbf{r} = \mathbf{0}$, where $A$ is the adjacency matrix of $G$ and the entries of $\mathbf{r}$ have no common factor other than $1$. In this paper, we generalize the result of Archer, Bishop, Diaz-Lopez, García Puente, Glass, and Louwsma on enumerating arithmetical structures on bidents (also called coconut tree graphs $\CT{p}{2}$) to all coconut tree graphs $\CT{p}{s}$ which consists of a path on $p>0$ vertices to which we append $s>0$ leaves to the right most vertex on the path. We also give a characterization of smooth arithmetical structures on coconut trees when given number assignments to the leaf nodes. |
| title | Arithmetical Structures on Coconut Trees |
| topic | Combinatorics 05C50, 05C30 |
| url | https://arxiv.org/abs/2406.11183 |