Carlos A. Alfaro

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.

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.

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|}

The Crossing Number of the Cone of a Graph

, let

Graphs with two trivial critical ideals

G^{\bf d}

Critical ideals, minimum rank and zero forcing number

be the graph obtained from

The equivalence between two classic algorithms for the assignment problem

G