# Arithmetical structures on bidents

## TL;DR

This paper investigates arithmetical structures on bidents, characterizing their critical groups and showing the number of such structures grows like Catalan numbers as the graph size increases.

## Contribution

It provides a method to determine the number of arithmetical structures on bidents and characterizes all possible critical groups for these structures.

## Key findings

- Number of arithmetical structures on bidents grows like Catalan numbers.
- Complete characterization of critical groups on bidents.
- A process for counting arithmetical structures on n-vertex bidents.

## Abstract

An arithmetical structure on a finite, connected graph $G$ is a pair of vectors $(\mathbf{d}, \mathbf{r})$ with positive integer entries for which $(\operatorname{diag}(\mathbf{d}) - A)\mathbf{r} = \mathbf{0}$, where $A$ is the adjacency matrix of $G$ and where the entries of $\mathbf{r}$ have no common factor. The critical group of an arithmetical structure is the torsion part of the cokernel of $(\operatorname{diag}(\mathbf{d}) - A)$. In this paper, we study arithmetical structures and their critical groups on bidents, which are graphs consisting of a path with two "prongs" at one end. We give a process for determining the number of arithmetical structures on the bident with $n$ vertices and show that this number grows at the same rate as the Catalan numbers as $n$ increases. We also completely characterize the groups that occur as critical groups of arithmetical structures on bidents.