Arithmetical Structures on Coconut Trees

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 $(\text{diag}(\mathbf{d})-A)\cdot \mathbf{r}=\mathbf{0}$, 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 $\text{CT}p2$) to all coconut tree graphs $\text{CT}ps$ 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.