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arXiv · 2406.11183

Arithmetical Structures on Coconut Trees

Alexander Diaz-Lopez·Brian Ha·Pamela E. Harris·Jonathan Rogers·Theo Koss·Dorian Smith
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Abstract

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, Garc\'ia 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.

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Alexander Diaz-Lopez, Brian Ha, Pamela E. Harris, Jonathan Rogers, Theo Koss, Dorian Smith. 2024-06-17. Arithmetical Structures on Coconut Trees. https://arxiv.org/abs/2406.11183

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