Arithmetical Structures on Coconut Trees

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.