论文标题
界定图上算术结构的数量
Bounding the number of arithmetical structures on graphs
论文作者
论文摘要
令$ g $为$ n $顶点的连接的无向图，没有循环，但可能是多层。给定算术结构$（\ textbf {r}，\ textbf {d}）$上的$ g $，我们描述了一种结构，该结构与$ n-1 $ pertices上的图$ g'$相关联，算术结构$（\ textbf {rtextbf {r}'，\ textbf {rextbf {d textbf {d d}'$ g'$ g'$ g'$ n $ g''）。通过迭代这种结构，我们仅根据$ g $的顶点和边缘的数量来得出$ g $的算术结构数量的上限。在完整图的特定情况下，可能具有多个边缘，我们将完善并将上限与计数单位分数表示形式产生的界限进行比较。
Let $G$ be a connected undirected graph on $n$ vertices with no loops but possibly multiedges. Given an arithmetical structure $(\textbf{r}, \textbf{d})$ on $G$, we describe a construction which associates to it a graph $G'$ on $n-1$ vertices and an arithmetical structure $(\textbf{r}', \textbf{d}')$ on $G'$. By iterating this construction, we derive an upper bound for the number of arithmetical structures on $G$ depending only on the number of vertices and edges of $G$. In the specific case of complete graphs, possibly with multiple edges, we refine and compare our upper bounds to those arising from counting unit fraction representations.
