TY - JOUR
T1 - On the critical group of matrices
AU - Corrales, Hugo
AU - Valencia, Carlos E.
PY - 2011
Y1 - 2011
N2 - Given a graph G with a distinguished vertex s, the critical group of (G, s) is the cokernel of their reduced Laplacian matrix L(G, s). In this article we generalize the concept of the critical group to the cokernel of any matrix with entries in a commutative ring with identity. In this article we find diagonal matrices that are equivalent to some matrices that generalize the reduced Laplacian matrix of the path, the cycle, and the complete graph over an arbitrary commutative ring with identity. We are mainly interested in those cases when the base ring is the ring of integers and some subrings of matrices. Using these equivalent diagonal matrices we calculate the critical group of the m-cones of the l-duplications of the path, the cycle, and the complete graph. Also, as byproduct, we calculate the critical group of another matrices, as the m-cones of the l-duplication of the bipartite complete graph with m vertices in each partition and the bipartite complete graph with 2m vertices minus a matching.
AB - Given a graph G with a distinguished vertex s, the critical group of (G, s) is the cokernel of their reduced Laplacian matrix L(G, s). In this article we generalize the concept of the critical group to the cokernel of any matrix with entries in a commutative ring with identity. In this article we find diagonal matrices that are equivalent to some matrices that generalize the reduced Laplacian matrix of the path, the cycle, and the complete graph over an arbitrary commutative ring with identity. We are mainly interested in those cases when the base ring is the ring of integers and some subrings of matrices. Using these equivalent diagonal matrices we calculate the critical group of the m-cones of the l-duplications of the path, the cycle, and the complete graph. Also, as byproduct, we calculate the critical group of another matrices, as the m-cones of the l-duplication of the bipartite complete graph with m vertices in each partition and the bipartite complete graph with 2m vertices minus a matching.
KW - Cartesian product
KW - Complete graph
KW - Critical group
KW - Cycle
KW - Matrices
KW - Path
UR - http://www.scopus.com/inward/record.url?scp=84871987293&partnerID=8YFLogxK
M3 - Artículo
SN - 1220-3874
VL - 54
SP - 213
EP - 236
JO - Bulletin Mathematique de la Societe des Sciences Mathematiques de Roumanie
JF - Bulletin Mathematique de la Societe des Sciences Mathematiques de Roumanie
IS - 3
ER -