This thesis is a contribution to the area of algebraic graph theory and the study of some generalizations of distance-regularity and related codes. In this work, the connection of graph theory and linear algebra is highlighted through the exhaustive use of the standard and local spectra of a graph, which is proven to be an appropriate tool to deal with such structures. The use of polynomials and, in particular, orthogonal polynomials of a discrete variable, is also ubiquitous. The relationship between the powers of the adjacency matrix and the distances in a graph make them suitable to deal with distance-related problems.
Distance-regular graphs have proved to be a key concept in algebraic com- binatorics. They have important connections with other branches of mathematics, such as geometry, coding theory, group theory, as well as with other areas of graph theory. This is because most finite objects bearing ¿enough regularity¿ are closely related to certain distance-regular graphs. The work initiated by Fiol, Garriga and Yebra in this area points out the relevance of orthogonal polynomials as a tool for the study of distanceregularity and some of its generalizations. Our first contribution is, in fact, to provide a common framework for the orthogonal polynomials used in the literature on this subject. Moreover, we also derive new applications for such polynomials and define some other families of polynomials that are suitable to deal with the studied properties.
One of our main objects of study are completely pseudo-regular codes, which generalize both local distance-regularity and completely regular codes. Our first approach to this notion comes from the study of the local spectra of a graph around the elements of a distance partition. We characterize these codes as those satisfying that the projections of the characteristic vectors of the elements of its distance partition are collinear. Inspired by the relation between the work of Terwilliger in the context of association schemes and the notion of local pseudo-distance-regularity, we also provide a new approach to completely pseudo-regular codes. Furthermore, we derive new characterizations which do not depend on the entire spectrum but only on the different eigenvalues in it and the characteristic vectors of the elements of the distance partition.
The study of completely pseudo-regular codes or, what is the same, pseudo-distance-regularity around a set, has the drawback that, in general, cannot be extended to a global point of view. However, by considering edges as sets of two vertices, we can define the notion of edge-distance-regularity and recover the global character inherent to distance-regularity. Many important problems that were conjectured or solved in the context of distance-regularity can also be posed for edge-distance-regular graphs. Through the use of an appropriate family of orthogonal polynomials, we give some analogues of some powerful results known for distance-regularity, such as the spectral excess theorem.