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.
Categories: math.CO math.NT