Arithmetical structures on graphs with connectivity one

Abstract

Given a graph [Formula: see text], an arithmetical structure on [Formula: see text] is a pair of positive integer vectors [Formula: see text] such that [Formula: see text] and [Formula: see text] where [Formula: see text] is the adjacency matrix of [Formula: see text]. We describe the arithmetical structures on graph [Formula: see text] with a cut vertex [Formula: see text] in terms of the arithmetical structures on their blocks. More precisely, if [Formula: see text] are the induced subgraphs of [Formula: see text] obtained from each of the connected components of [Formula: see text] by adding the vertex [Formula: see text] and their incident edges, then the arithmetical structures on [Formula: see text] are in one to one correspondence with the [Formula: see text]-rational arithmetical structures on the [Formula: see text]’s. A rational arithmetical structure corresponds to an arithmetical structure where some of the integrality conditions are relaxed.