TY - JOUR

T1 - Arithmetical structures on graphs with connectivity one

AU - Corrales, Hugo

AU - Valencia, Carlos E.

PY - 2018/8/1

Y1 - 2018/8/1

N2 - © 2018 World Scientific Publishing Company. 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.

AB - © 2018 World Scientific Publishing Company. 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.

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U2 - 10.1142/S0219498818501475

DO - 10.1142/S0219498818501475

M3 - Article

JO - Journal of Algebra and its Applications

JF - Journal of Algebra and its Applications

SN - 0219-4988

ER -