Arithmetical structures on graphs with connectivity one

Given a graph G, an arithmetical structure on G is a pair of positive integer vectors (d,r) such that gcd(rv|v V (G)) = 1 and (diag(d)-A)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 G1,..,Gs 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 Gi's. A rational arithmetical structure corresponds to an arithmetical structure where some of the integrality conditions are relaxed.