Resumen de Contribución al estudio de excentricidades en grupos dirigidos
Graphs are used as mathematical models for all kind of networks, Most of the basic properties concerning optimal performance of such networks are related to distance in graphs. These properties include the distance from a particular vertex to the farthest vertex of that one, that is, the eccentricity. The metric properties of graphs, both directed and undirected, and particularly the study of eccentricities, is the main purpose of this thesis. We start our work with an approach to metric operators related with distance, specially the eccentric digraph operator. Among others, we consider the symmetry problem and the characterization of eccentric digraphs. The chapter ends laying the basis of a generalized metric operator that attempts to unify every distance-related operator. The third chapter is devoted to iterated sequences of eccentric digraphs. We expose the study of known parameters that have relations with such sequences. In particular, we show that they depend, in some cases, on the connectivity in what we call the eccentric core. The fourth chapter deals with eccentricity sequences of graphs and digraphs. While eccentricity sequences of graphs have been studied by several authors, we explore the directed case 'adapting' new results for digraphs, we show the eccentric character of some 'extremal' sequences and we completely characterize the sets formed by eccentricities in digraphs. The final part of our work is devoted to the study of an 'eccentric' variation of Moore digraphs: radially Moore digraphs. For them, we create an operator based on the line digraph which allows us to construct new radially Moore digraphs. Moreover, the unique known family of radially Moore digraphs can be seen as a particular case of application of this operator. In the directed case, we give a general construction of a radially Moore graph of radius two and we present different ways that it can be insightful for the general case.
