arXiv subjects

Carlos E. Valencia

Publications and source records attributed to Carlos E. Valencia.

22 records · Page 2Linked to original sources

arXiv2011

On the Sandpile group of the cone of a graph

In this article, we give a partial description of the sandpile group of the cone of the cartesian product of graphs in function of the sandpile group of the cone of their factors. Also, we introduce the concept of uniform homomorphism of graphs and prove that every surjective uniform homomorphism of graphs induces an injective homomorphism between their sandpile groups. As an application of these result we obtain an explicit description of a set of generators of the sandpile group of the cone of the hypercube of dimension d.

math.CO↗

arXiv2010

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.

math.AC↗

arXiv2007

1-join composition for $α$-critical graphs

Given two graphs G and H its 1-{\it join} is the graph obtained by taking the disjoint union of G and H and adding all the edges between a nonempty subset of vertices of G and a nonempty subset of vertices of H. In general, composition operations of graphs has played a fundamental role in some structural results of graph theory and in particular the 1-join composition has played an important role in decomposition theorems of several class of graphs such as the claw-free graphs, the bull-free graphs, the perfect graphs, etc. A graph G is called {\it $α$-critical} if $α(G\setminus e)> α(G)$ for all the edges e of G, where $α(G)$, the {\it stability number} of G, is equal to the maximum cardinality of a stable set of G, and a set of vertices M of G is {\it stable} if no two vertices in M are adjacent. The study $α$-critical graphs is important, for instance a complete description of $α$-critical graphs would yield a good characterization of the stability number of G. In this paper we give necessary and sufficient conditions that G and H must satisfy in order to its 1-join will be an $α$-critical graph. Therefore we get a very useful way to construct basic $α$-critical graphs using the 1-join of graphs.

math.CO↗

arXiv2005

Explicit representations by halfspaces of the edge cone of a graph

Let G be an arbitrary simple graph. The main results are explicit representations of the edge cone of G as a finite intersection of closed halfspaces. If G is bipartite and connected we determine the facets of the edge cone and present a canonical irreducible representation.

math.CO↗