Carlos E. Valencia

On the sandpile group of the cone of a graph

Abstract In this article, we study the sandpile group of the cone of a graph. After introducing the concept of uniform homomorphism of graphs we prove that every surjective uniform homomorphism of graphs induces an injective homomorphism between their sandpile groups. Also, we establish a relationship between the sandpile group of the cone of the cartesian product of graphs and the sandpile group of the cone of their factors. As an application of these results we obtain an explicit description of a set of generators of the sandpile group of the cone of the hypercube.

Canonical modules of certain edge subrings

Let G be a connected bipartite graph. We present an approach to the computation of the canonical module of the edge subring associated to G using linear programming.

Dimension reduction in principal component analysis for trees

The statistical analysis of tree structured data is a new topic in statistics with wide application areas. Some Principal Component Analysis (PCA) ideas have been previously developed for binary tree spaces. These ideas are extended to the more general space of rooted and ordered trees. Concepts such as tree-line and forward principal component tree-line are redefined for this more general space, and the optimal algorithm that finds them is generalized. An analog of the classical dimension reduction technique in PCA for tree spaces is developed. To do this, backward principal components, the components that carry the least amount of information on tree data set, are defined. An optimal algorithm to find them is presented. Furthermore, the relationship of these to the forward principal components is investigated, and a path-independence property between the forward and backward techniques is proven. These methods are applied to a brain artery data set of 98 subjects. Using these techniques, the effects of aging to the brain artery structure of males and females is investigated. A second data set of the organization structure of a large US company is also analyzed and the structural differences across different types of departments within the company are explored.

On the critical ideals of graphs

We introduce some determinantal ideals of the generalized Laplacian matrix associated to a digraph G, that we call critical ideals of G. Critical ideals generalize the critical group and the characteristic polynomials of the adjacency and Laplacian matrices of a digraph. The main results of this article are the determination of some minimal generator sets and the reduced Grobner basis for the critical ideals of the complete graphs, the cycles and the paths. Also, we establish a bound between the number of trivial critical ideals and the stability and clique numbers of a graph.

Critical ideals of signed graphs with twin vertices

This paper studies critical ideals of graphs with twin vertices, which are vertices with the same neighbors. A pair of such vertices are called replicated if they are adjacent, and duplicated, otherwise. Critical ideals of graphs having twin vertices have good properties and show regular patterns. Given a graph

Small clique number graphs with three trivial critical ideals

G=(V,E)

Digraphs with at most one trivial critical ideal

and

Graphs with few trivial critical ideals

{\bf d}\in \mathbb{Z}^{|V|}

Arithmetical structures on graphs with connectivity one

, let

Counting arithmetical structures on paths and cycles

G^{\bf d}