Claudia Justel

Subclasses of k-trees: Characterization and recognition

A k-tree is either a complete graph on k vertices or a graph G = (V, E) that contains a vertex whose neighbourhood in G induces a complete graph on k vertices and whose removal results in a k-tree. We present two new subclasses of k-trees and their properties. First, we present the definition and characterization of k-path graphs, based on the concept of k-paths, that generalizes the classic concept of paths. We also introduce the simple-clique k-trees, of which the maximal outerplanar graphs and the planar 3-trees are particular cases. Based on Characterization Theorems, we show recognition algorithms for both families. Finally, we establish the inclusion relations among these new classes and k-trees.

EXTREMAL ALGEBRAIC CONNECTIVITIES OF CERTAIN CATERPILLAR CLASSES AND SYMMETRIC CATERPILLARS

A caterpillar is a tree in which the removal of all pendant vertices makes it a path. Let d ≥ 3 and n ≥ 6 be given. Let Pd 1 be the path of d − 1 vertices and Sp be the star of p + 1 vertices. Let p = (p1,p2,...,pd 1) such that p1 ≥ 1,p2 ≥ 1,...,pd 1 ≥ 1. Let C (p) be the caterpillar obtained from the stars Sp1,Sp2,...,Spd 1 and the path Pd 1 by identifying the root of Spi with the i−vertex of Pd 1. Let n > 2(d − 1) be given. Let In this paper, the caterpillars in C and in S having the maximum and the minimum algebraic connectivity are found. Moreover, the algebraic connectivity of a caterpillar in S as the smallest eigenvalue of a 2 × 2 - block tridiagonal matrix of order 2s × 2s if d = 2s + 1 or d = 2s + 2 is characterized.

Experiments with two heuristic algorithms for the Maximum Algebraic Connectivity Augmentation Problem

Abstract In this work we present a heuristic algorithm to solve the Maximum Algebraic Connectivity Augmentation Problem (MACAP). This is an NP-complete problem (proved by Mosk-Aoyama in 2008) and consists in, given a graph, determining the smallest set of edges not belonging to it in such a way that the value of the algebraic connectivity of the augmented graph is maximum. In 2006, Ghosh and Boyd presented a heuristic procedure to solve this problem. This heuristic is an iterative method that selects one edge at a time based on the values of the components of a Fiedler vector of the graph. Our goal is to increase the value of the algebraic connectivity of a given graph by inserting edges based on the eccentricity of vertices. In order to evaluate our algorithm, computational tests comparing it with the Ghosh and Boyd procedure are presented.

Visual similarity analysis in loop closure through data dimensionality reduction via diffusion maps

This paper describes loop closures detection, a significant problem in mobile robotics, using analysis of similarity between images in a low-dimensional mapping. We represent a set of images as a graph in high-dimensional space, where each node is represented by a dominant eigenvector of the correspondent image. To this graph, we apply Diffusion Maps by Coifman and Lafon [4], a graph-based spectral method to data dimensionality reduction. We determine visual similarity analysis and detect loop closure in lower dimension, without building a vocabulary of visual words. Our experiments show results of loop closures detection both in indoor and outdoor environments from images captured by an RGB camera as well as images captured using Google Street View.

Measures of irregularity of graphs

A graph is regular if every vertex is of the same degree. Otherwise, it is an irregular graph. Although there is a vast literature devoted to regular graphs, only a few papers approach the irregular ones. We have found four distinct graph invariants used to measure the irregularity of a graph. All of them are determined through either the average or the variance of the vertex degrees. Among them there is the index of the graph, a spectral parameter, which is given as a function of the maximum eigenvalue of its adjacency matrix. In this paper, we survey these invariants with highlight to their respective properties, especially those relative to extremal graphs. Finally, we determine the maximum values of those measures and characterize their extremal graphs in some special classes.

Maximizing the algebraic connectivity for a subclass of caterpillars

Abstract A caterpillar is a tree in which the removal of all pendant vertices makes it a path. Let d ⩾ 3 and n ⩾ 6 be given. Let P d − 1 be the path on d − 1 vertices and K 1 , p be the star of p + 1 vertices. Let p = [ p 1 , p 2 , … , p d − 1 ] such that ∀ i , 1 ⩽ i ⩽ d − 1 , p i . Let C ( p ) be the caterpillar obtained from d − 1 stars K 1 , p i and the path P d − 1 by identifying the root of K 1 , p i with the i-vertex of P d − 1 . For a given n ⩾ 2 ( d − 1 ) , let C = { C ( p ) : ∑ i = 1 , d − 1 p i = n − d + 1 } . In this work, we give the caterpillar in C maximizing the algebraic connectivity.

Managing Graph Modeling Alternatives for Link Prediction.

The importance of bringing the relational data to other models and technologies has been widely debated, as for example their representation as graphs. This model allows to perform topological analysis such as social analysis, link predictions or recommendations. There are already initiatives to map from a relational database to graph representation. However, they do not take into account the different ways to generate such graphs from data stored in relational databases, specially when the goal is to perform topological analysis. This work discusses how graph modeling alternatives from data stored in relational datasets may lead to useful results. However, this is not an easy task. The main contribution of this paper is towards managing such alternatives, taking into account that the graph model choice and the topological analysis to be used, depend on the links the user intends to predict. Experiments are reported and show interesting results, including modeling heuristics to guide the user on the graph model choice.

Augmenting the algebraic connectivity for certain families of graphs

Abstract In this work we compare solutions of two distinct algorithms that try to increase the value of the algebraic connectivity of a given graph by choosing, with different strategies, edges to be included. Eccentricity and Perturbation Heuristics were considered, as well as random graphs and special families of trees being inputs of those algorithms. As conclusions of the reported experiments, the Eccentricity Heuristic obtains good results when compared with Perturbation Heuristic and a conjecture about broom trees is presented.

Experimental implementation of loop closure detection using data dimensionality reduction by spectral method

This paper presents experimental results about loop closure detection in mobile robots through spectral description of images set and data dimensionality reduction. Both, spectral description and representation in low dimension depend heavily on the concept of dominant eigenvector. Integration between Matlab and ROS interface was exploited to perform our experiments. Besides, two environments were used: real and computationally simulated. Results have shown that the method is capable of performing correct loop closure detection at a significantly lower computation cost, when compared with those obtained by OpenCV library for visual analysis.

On the characteristic set, centroid, and centre for a tree

Abstract Given a tree T, we consider a pair of vertices (u, v) where u is a centroid of T, v is a characteristic vertex of T, and such that the distance between them, denoted d(u, v), is smallest over all such pairs. We define and where the maximum is taken over all trees T on n vertices. Analogous definitions are also given for and . We show that for each there is a broom T on n vertices such that and a broom on n vertices such that We also prove that the sequences and are convergent, and find their limits. We rely on the characterization of characteristic vertices in terms of Perron branches in order to establish our results.