2016/06/12 by Corrales, Hugo, Valencia, Carlos E. · 1 citation
#11C20 #11D72. (Secondary) #14C17 #15B36 (Primary) #Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT)
paper · doi:10.48550/arxiv.1606.03726
Given a graph G, an arithmetical structure on G is a pair of positive integer vectors (\bf d,\bf r) such that gcd(\bf rv | v∈ V(G))=1 and (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 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. We introduce the concept of rational arithmetical structure, which corresponds to an arithmetical structure where some of the integrality conditions are relaxed.