Arithmetical structures of graphs II: Graphs with connectivity one

Hugo Corrales, Carlos E. Valencia

Published 2016-06-12Version 1

We give a description of the arithmetical structures of a graph G with a cut vertex v in function of the arithmetical structures of its blocks. More precisely, if G_1,...,G_s are the induced subgraphs of G obtained from each of the connected components of G-v by adding the vertex v and its incident edges, then the arithmetical structures of G are in correspondence with the v-rational arithmetical structures of the G_i's. We introduce a key concept, of rational arithmetical structure, which correspond to an arithmetical structure were some integer conditions are relaxed. This concept is useful in a wide number of situations, as in the case of graphs with twin vertices.