TY - JOUR
T1 - Arithmetical structures on graphs with connectivity one
AU - Corrales, Hugo
AU - Valencia, Carlos E.
N1 - Publisher Copyright:
© 2018 World Scientific Publishing Company.
PY - 2018/8/1
Y1 - 2018/8/1
N2 - 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 - 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.
KW - 2-connected components
KW - Arithmetical graphs
KW - Laplacian matrix
KW - M-matrices
KW - arithmetical structure
KW - connectivity one
KW - cut vertex
UR - http://www.scopus.com/inward/record.url?scp=85027681184&partnerID=8YFLogxK
U2 - 10.1142/S0219498818501475
DO - 10.1142/S0219498818501475
M3 - Artículo
SN - 0219-4988
VL - 17
JO - Journal of Algebra and its Applications
JF - Journal of Algebra and its Applications
IS - 8
M1 - 1850147
ER -