# Arithmetical structures on graphs

• Published in 2016
In the collection
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.

## Other information

key
Arithmeticalstructuresongraphs
type
article
2017-06-14
date_published
2016-09-05

### BibTeX entry

@article{Arithmeticalstructuresongraphs,
key = {Arithmeticalstructuresongraphs},
type = {article},
title = {Arithmetical structures on graphs},
author = {Hugo Corrales and Carlos E. Valencia},
abstract = {Arithmetical structures on a graph were introduced by Lorenzini as some
intersection matrices that arise in the study of degenerating curves 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.},
comment = {},
}