Arithmetical structures on graphs

TL;DR

This paper explores arithmetical structures on graphs, focusing on complete graphs, paths, and cycles, providing explicit descriptions and counting results, including a Catalan number enumeration for paths.

Contribution

It characterizes arithmetical structures on various graph classes and introduces new descriptions for structures on modified graphs, extending previous results.

Findings

Number of arithmetical structures on a path equals a Catalan number.

Provides explicit descriptions of arithmetical structures on paths and cycles.

Shows finiteness of arithmetical structures on any graph.

Abstract

Arithmetical structures on a graph were introduced by Lorenzini as some intersection matrices that arise in the study of degenerating curves in algebraic geometry. In this article we study these arithmetical structures, in particular we are interested in the arithmetical structures on complete graphs, paths, and cycles. We begin by looking at the arithmetical structures on a multidigraph from the general perspective of $M$-matrices. As an application, we recover the result of Lorenzini about the finiteness of the number of arithmetical structures on a graph. We give a description on the arithmetical structures on the graph obtained by merging and splitting a vertex of a graph in terms of its arithmetical structures. On the other hand, we give a description of the arithmetical structures on the clique--star transform of a graph, which generalizes the subdivision of a graph. As an application of this result we obtain an explicit description of all the arithmetical structures on the paths and cycles and we show that the number of the arithmetical structures on a path is a Catalan number.