  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.
