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 (r,d) on G, we describe a construction which associates to it a graph G′ on n−1 vertices and an arithmetical structure (r′,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...