Bounding the number of arithmetical structures on graphs

Abstract

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.