Convex extendable trees

The concept of convex extendability is introduced to answer the problem of finding the smallest distance convex simple graph containing a given tree. A problem of similar type with respect to minimal path convexity is also discussed.

On Distance Energy of Graphs

The D-eigenvalues of a graph G are the eigenvalues of its distance matrix D, and the D-energy ED(G) is the sum of the absolute values of its D-eigenvalues. Two graphs are said to be D-equienergetic if they have the same D-energy. In this note we obtain bounds for the distance spectral radius and D-energy of graphs of diameter 2. Pairs of equiregular D-equienergetic graphs of diameter 2, on p = 3t + 1 vertices are also constructed.

ON DISTANCE ENERGY OF GRAPHS

The D-eigenvalues of a graph G are the eigenvalues of its distance matrix D, and the D-energy ED(G) is the sum of the absolute values of its D-eigenvalues. Two graphs are said to be D-equienergetic if they have the same D-energy. In this note we obtain bounds for the distance spectral radius and D-energy of graphs of diameter 2. Pairs of equiregular D-equienergetic graphs of diameter 2, on p = 3t + 1 vertices are also constructed.

Algorithms for Maximum Independent Set in Convex Bipartite Graphs

A bipartite graph G = (V, W, E) is convex if there exists an ordering of the vertices of W such that, for each v. V, the neighbors of v are consecutive in W. We describe both a sequential and a BSP/CGM algorithm to find a maximum independent set in a convex bipartite graph. The sequential algorithm improves over the running time of the previously known algorithm and the BSP/CGM algorithm is a parallel version of the sequential one. The complexity of the algorithms does not depend on |W|.

A educação a distância e a formação continuada de professores de matemática: contribuições de um contexto formativo para a base de conhecimento docente

Pós-graduação em Educação - FCT

GENEOs, compactifications, and graphs

Our objective in this thesis is to study the pseudo-metric and topological structure of the space of group equivariant non-expansive operators (GENEOs). We introduce the notions of compactification of a perception pair, collectionwise surjectivity, and compactification of a space of GENEOs. We obtain some compactification results for perception pairs and the space of GENEOs. We show that when the data spaces are totally bounded and endow the common domains with metric structures, the perception pairs and every collectionwise surjective space of GENEOs can be embedded isometrically into the compact ones through compatible embeddings. An important part of the study of topology of the space of GENEOs is to populate it in a rich manner. We introduce the notion of a generalized permutant and show that this concept too, like that of a permutant, is useful in defining new GENEOs. We define the analogues of some of the aforementioned concepts in a graph theoretic setting, enabling us to use the power of the theory of GENEOs for the study of graphs in an efficient way. We define the notions of a graph perception pair, graph permutant, and a graph GENEO. We develop two models for the theory of graph GENEOs. The first model addresses the case of graphs having weights assigned to their vertices, while the second one addresses weighted on the edges. We prove some new results in the proposed theory of graph GENEOs and exhibit the power of our models by describing their applications to the structural study of simple graphs. We introduce the concept of a graph permutant and show that this concept can be used to define new graph GENEOs between distinct graph perception pairs, thereby enabling us to populate the space of graph GENEOs in a rich manner and shed more light on its structure.

Matlab self study tutorial - Graphing

These resources are designed to support students in gaining more confidence with using Matlab. The PDFs provide guidance and information; Objectives: Introduce basic syntax and data preparation for graphing with Matlab by providing some data, examples of code and some background documents. Outcomes: -how to write an m file script -the importance of syntax -how to load files -how to produce simple graphs -where to get help and further examples There are also some data files to provide example data for students to work with in producing Matlab resources.

Weighted alpha-rate dominating sets in social networks

We are looking into variants of a domination set problem in social networks. While randomised algorithms for solving the minimum weighted domination set problem and the minimum alpha and alpha-rate domination problem on simple graphs are already present in the literature, we propose here a randomised algorithm for the minimum weighted alpha-rate domination set problem which is, to the best of our knowledge, the first such algorithm. A theoretical approximation bound based on a simple randomised rounding technique is given. The algorithm is implemented in Python and applied to a UK Twitter mentions networks using a measure of individuals’ influence (klout) as weights. We argue that the weights of vertices could be interpreted as the costs of getting those individuals on board for a campaign or a behaviour change intervention. The minimum weighted alpha-rate dominating set problem can therefore be seen as finding a set that minimises the total cost and each individual in a network has at least alpha percentage of its neighbours in the chosen set. We also test our algorithm on generated graphs with several thousand vertices and edges. Our results on this real-life Twitter networks and generated graphs show that the implementation is reasonably efficient and thus can be used for real-life applications when creating social network based interventions, designing social media campaigns and potentially improving users’ social media experience.

Effects of shape Rosmarinus officinalis neighbors on resprouting of shape Erica multiflora individuals

To study the effects of competition in Mediterranean shrubland regeneration following disturbance, we used a neighborhood approach to assess the influence of mature Rosmarinus officinalis neighbors on the resprouting of Erica multiflora individuals after clipping. Sprout biomass of target plants 2 years after clipping was regressed against various measures of neighbor abundance within a 2 m radius around target E. multiflora individuals in which all vegetation except R. officinalis had been removed. The largest single influence on the biomass of sprouts produced was the previous biomass of the resprouting plant. The abundance of R. officinalis neighbors had a weak but detectable effect on resprouting of E. multiflora. Abundance of neighbors within 60 cm from target plants was the best predictor of regrowth. At this distance, two simple measures of neighbor abundance within the neighborhood, the number of neighbors and the sum of their heights, were significant in accounting for variation in resprouted biomass. None of the combinations of neighbor variables performed significantly better than single variables. The best models accounted for around 24 percent of the variation in resprout biomass. As in other studies, angular dispersion of neighbors never had a significant effect on performance of target plants. The weak but significant response of resprouting to variation in R. officinalis abundance suggests that the intensity of competition in the experiment was low because of the removal of other species.

Medida de distancia semántica en grafos UNL

El trabajo que se presenta a continuación desarrolla un modelo para calcular la distancia semántica entre dos oraciones representadas por grafos UNL. Este problema se plantea en el contexto de la traducción automática donde diferentes traductores pueden generar oraciones ligeramente diferentes partiendo del mismo original. La medida de distancia que se propone tiene como objetivo proporcionar una evaluación objetiva sobre la calidad del proceso de generación del texto. El autor realiza una exploración del estado del arte sobre esta materia, reuniendo en un único trabajo los modelos propuestos de distancia semántica entre conceptos, los modelos de comparación de grafos y las pocas propuestas realizadas para calcular distancias entre grafos conceptuales. También evalúa los pocos recursos disponibles para poder experimentar el modelo y plantea una metodología para generar los conjuntos de datos que permitirían aplicar la propuesta con el rigor científico necesario y desarrollar la experimentación. Utilizando las piezas anteriores se propone un modelo novedoso de comparación entre grafos conceptuales que permite utilizar diferentes algoritmos de distancia entre conceptos y establecer umbrales de tolerancia para permitir una comparación flexible entre las oraciones. Este modelo se programa utilizando C++, se alimenta con los recursos a los que se ha hecho referencia anteriormente, y se experimenta con un conjunto de oraciones creado por el autor ante la falta de otros recursos disponibles. Los resultados del modelo muestran que la metodología y la implementación pueden conducir a la obtención de una medida de distancia entre grafos UNL con aplicación en sistemas de traducción automática, sin embargo, la carencia de recursos y de datos etiquetados con los que validar el algoritmo requieren un esfuerzo previo importante antes de poder ofrecer resultados concluyentes.---ABSTRACT---The work presented here develops a model to calculate the semantic distance between two sentences represented by their UNL graphs. This problem arises in the context of machine translation where different translators can generate slightly different sentences from the same original. The distance measure that is proposed aims to provide an objective evaluation on the quality of the process involved in the generation of text. The author carries out an exploration of the state of the art on this subject, bringing together in a single work the proposed models of semantic distance between concepts, models for comparison of graphs and the few proposals made to calculate distances between conceptual graphs. It also assesses the few resources available to experience the model and presents a methodology to generate the datasets that would be needed to develop the proposal with the scientific rigor required and to carry out the experimentation. Using the previous parts a new model is proposed to compute differences between conceptual graphs; this model allows the use of different algorithms of distance between concepts and is parametrized in order to be able to perform a flexible comparison between the resulting sentences. This model is implemented in C++ programming language, it is powered with the resources referenced above and is experienced with a set of sentences created by the author due to the lack of other available resources. The results of the model show that the methodology and the implementation can lead to the achievement of a measure of distance between UNL graphs with application in machine translation systems, however, lack of resources and of labeled data to validate the algorithm requires an important effort to be done first in order to be able to provide conclusive results.

On Graph-Based Cryptography and Symbolic Computations

We have been investigating the cryptographical properties of in nite families of simple graphs of large girth with the special colouring of vertices during the last 10 years. Such families can be used for the development of cryptographical algorithms (on symmetric or public key modes) and turbocodes in error correction theory. Only few families of simple graphs of large unbounded girth and arbitrarily large degree are known. The paper is devoted to the more general theory of directed graphs of large girth and their cryptographical applications. It contains new explicit algebraic constructions of in finite families of such graphs. We show that they can be used for the implementation of secure and very fast symmetric encryption algorithms. The symbolic computations technique allow us to create a public key mode for the encryption scheme based on algebraic graphs.

Leaving and Coming Back to Canada : Evidence from Longitudinal Data (Working Paper 32)

The paper exploits the unique strengths of Statistics Canada's Longitudinal Administrative Database ("LAD"), constructed from individuals' tax records, to shed new light on the extent and nature of the emigration of Canadians to other countries and their patterns of return over the period 1982-1999. The empirical evidence begins with some simple graphs of the overall rates of leaving over time, and follows with the presentation of the estimation results of a model that essentially addresses the question: "who moves?" The paper then analyses the rates of return for those observed to leave the country - something for which there is virtually no existing evidence. Simple return rates are reported first, followed by the results of a hazard model of the probability of returning which takes into account individuals' characteristics and the number of years they have already been out of the country. Taken together, these results provide a new empirical basis for discussions of emigration in general, and the brain drain in particular. Of particular interest are the ebb and flow of emigration rates observed over the last two decades, including a perhaps surprising turndown in the most recent years after climbing through the earlier part of the 1990s; the data on the number who return after leaving, the associated patterns by income level, and the increases observed over the last decade.

Packings and coverings of the complete graph with trees

In this thesis, we define the spectrum problem for packings (coverings) of G to be the problem of finding all graphs H such that a maximum G-packing (minimum G- covering) of the complete graph with the leave (excess) graph H exists. The set of achievable leave (excess) graphs in G-packings (G-coverings) of the complete graph is called the spectrum of leave (excess) graphs for G. Then, we consider this problem for trees with up to five edges. We will prove that for any tree T with up to five edges, if the leave graph in a maximum T-packing of the complete graph Kn has i edges, then the spectrum of leave graphs for T is the set of all simple graphs with i edges. In fact, for these T and i and H any simple graph with i edges, we will construct a maximum T-packing of Kn with the leave graph H. We will also show that for any tree T with k ≤ 5 edges, if the excess graph in a minimum T-covering of the complete graph Kn has i edges, then the spectrum of excess graphs for T is the set of all simple graphs and multigraphs with i edges, except for the case that T is a 5-star, for which the graph formed by four multiple edges is not achievable when n = 12.