Arithmetical Structures on Coconut Trees

TL;DR

This paper extends the enumeration of arithmetical structures from specific coconut tree graphs to a broader class, providing a comprehensive characterization of these structures on all coconut trees.

Contribution

It generalizes previous enumeration results to all coconut tree graphs and characterizes smooth arithmetical structures with leaf node assignments.

Findings

Enumeration of arithmetical structures on all coconut trees

Characterization of smooth arithmetical structures

Extension of previous results on bidents

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.