11,231 research outputs found
On the critical group of matrices
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, the bipartite complete graph with 2m vertices minus a matching.Comment: 18 pages, 5 figure
Optimum matchings in weighted bipartite graphs
Given an integer weighted bipartite graph we consider the problems of finding all the edges that occur in
some minimum weight matching of maximum cardinality and enumerating all the
minimum weight perfect matchings. Moreover, we construct a subgraph of
which depends on an -optimal solution of the dual linear program
associated to the assignment problem on that allows us to reduced
this problems to their unweighed variants on . For instance, when
has a perfect matching and we have an -optimal solution of the dual
linear program associated to the assignment problem on , we solve the
problem of finding all the edges that occur in some minimum weight perfect
matching in linear time on the number of edges. Therefore, starting from
scratch we get an algorithm that solves this problem in time
, where , , and .Comment: 11 page
Small clique number graphs with three trivial critical ideals
The critical ideals of a graph are the determinantal ideals of the
generalized Laplacian matrix associated to a graph. In this article we provide
a set of minimal forbidden graphs for the set of graphs with at most three
trivial critical ideals. Then we use these forbidden graphs to characterize the
graphs with at most three trivial critical ideals and clique number equal to 2
and 3.Comment: 33 pages, 3 figure
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