Arithmetical structures on graphs with connectivity one

TL;DR

This paper characterizes arithmetical structures on graphs with a cut vertex by relating them to structures on their blocks, introducing the concept of rational arithmetical structures to generalize the framework.

Contribution

It provides a decomposition method for arithmetical structures on graphs with connectivity one and introduces rational arithmetical structures as a new concept.

Findings

Arithmetical structures on graphs with a cut vertex are in one-to-one correspondence with v-rational structures on blocks.

The concept of rational arithmetical structures generalizes traditional arithmetical structures.

The paper offers a systematic way to analyze arithmetical structures via graph decomposition.

Abstract

Given a graph $G$, an arithmetical structure on $G$ is a pair of positive integer vectors $({\bf d},{\bf r})$ such that $\mathrm{gcd}({\bf r}_v\, | \,v\in V(G))=1$ and \[ (\mathrm{diag}({\bf d})-A){\bf r}=0, \] where $A$ is the adjacency matrix of $G$. We describe the arithmetical structures on graph $G$ with a cut vertex $v$ in terms of the arithmetical structures on their blocks. More precisely, if $G_1,\ldots,G_s$ are the induced subgraphs of $G$ obtained from each of the connected components of $G-v$ by adding the vertex $v$ and their incident edges, then the arithmetical structures on $G$ are in one to one correspondence with the $v$-rational arithmetical structures on the $G_i$'s. We introduce the concept of rational arithmetical structure, which corresponds to an arithmetical structure where some of the integrality conditions are relaxed.