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 \((\diag(\mathbf{d}) - A)\cdot \mathbf{r} = \mathbf{0}\), 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.