Bounding the number of arithmetical structures on graphs

TL;DR

This paper introduces a method to bound the number of arithmetical structures on graphs by constructing related graphs and iterating this process, with specific bounds for complete graphs and comparisons to unit fraction counts.

Contribution

It presents a novel construction linking arithmetical structures on a graph to those on smaller graphs, enabling upper bounds based on graph parameters.

Findings

Derived an upper bound for arithmetical structures depending on vertices and edges.

Refined bounds for complete graphs with multiple edges.

Compared bounds to counts of unit fraction representations.

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.