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.

中文翻译：

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bidents 上的算术结构

有限连通图 $G$ 上的算术结构是一对向量 $(\mathbf{d}, \mathbf{r})$ 具有正整数项，其中 $(\operatorname{diag}(\mathbf{d} }) - A)\mathbf{r} = \mathbf{0}$，其中$A$ 是$G$ 的邻接矩阵，$\mathbf{r}$ 的条目没有公因子。算术结构的临界群是 $(\operatorname{diag}(\mathbf{d}) - A)$ 的 cokernel 的扭转部分。在本文中，我们研究了算术结构及其对双联的临界群，双联是由一端有两个“尖头”的路径组成的图。我们给出了一个确定具有 $n$ 顶点的二元算术结构数量的过程，并表明随着 $n$ 的增加，这个数字与加泰罗尼亚数字的增长速度相同。