论文标题
在广义恒星运行下的算术结构的临界组
Critical groups of arithmetical structures under a generalized star-clique operation
论文作者
论文摘要
在有限的，连接的图形上的算术结构是由向顶点的正整数分配给出的正整数，即在每个顶点，整数在相邻顶点处有一个整数的总和，如果图形并不简单，则在相邻的顶点处有多个。与每个算术结构相关的是一个被称为其关键组的有限阿贝尔群。凯斯（Keyes）和雷特（Reiter）进行了一个操作，该操作在没有循环的有限，连接的图形上进行了算术结构，并在图表上产生算术结构，其顶点较少。我们研究该操作如何改变关键群体。我们绑定了最初的算术结构和关键组的临界组的顺序和不变因素。当原始图很简单时，我们会准确确定所得关键组。
An arithmetical structure on a finite, connected graph without loops is given by an assignment of positive integers to the vertices such that, at each vertex, the integer there is a divisor of the sum of the integers at adjacent vertices, counted with multiplicity if the graph is not simple. Associated to each arithmetical structure is a finite abelian group known as its critical group. Keyes and Reiter gave an operation that takes in an arithmetical structure on a finite, connected graph without loops and produces an arithmetical structure on a graph with one fewer vertex. We study how this operation transforms critical groups. We bound the order and the invariant factors of the resulting critical group in terms of the original arithmetical structure and critical group. When the original graph is simple, we determine the resulting critical group exactly.
