Bounding the number of arithmetical structures on graphs

Let $G$ be a connected undirected graph on $n$ vertices with no loops but possibly multiedges. Given an arithmetical structure $(\mathbf{\text{r}},\mathbf{\text{d}})$ on $G$, we describe a construction which associates to it a graph $G^{\mathrm{\prime }}$ on $n-1$ vertices and an arithmetical structure $({\mathbf{\text{r}}}^{\mathrm{\prime }},{\mathbf{\text{d}}}^{\mathrm{\prime }})$ on $G^{\mathrm{\prime }}$. 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.