Some applications of linear algebra in spectral graph theory

The application of the theory of matrices and eigenvalues to combinatorics is cer- tainly not new. In the present work the starting point is a theorem that concerns the eigenvalues of partitioned matrices. Interlacing yields information on subgraphs of a graph, and the way such subgraphs are embedded. In particular, one gets bounds on extremal substructures. Applications of this theorem and of some known matrix theorems to matrices associated to graphs lead to new results. For instance, some characterizations of regular partitions, and bounds for some parameters, such as the independence and chromatic numbers, the diameter, the bandwidth, etc. This master thesis is a contribution to the area of algebraic graph theory and the study of some generalizations of regularity in bipartite graphs. In Chapter 1 we recall some basic concepts and results from graph theory and linear algebra. Chapter 2 presents some simple but relevant results on graph spectra concerning eigenvalue interlacing. Most of the previous results that we use were obtained by Haemers in [33]. In that work, the author gives bounds for the size of a maximal (co)clique, the chromatic number, the diameter and the bandwidth in terms of the eigenvalues of the standard adjacency matrix or the Laplacian matrix. He also nds some inequalities and regularity results concerning the structure of graphs. The work initiated by Fiol [26] in this area leads us to Chapter 3. The discussion goes along the same spirit, but in this case eigenvalue interlacing is used for proving results about some weight parameters and weight-regular partitions of a graph. In this master thesis a new observation leads to a greatly simpli ed notation of the results related with weight-partitions. We nd an upper bound for the weight independence number in terms of the minimum degree. Special attention is given to regular bipartite graphs, in fact, in Chapter 4 we contribute with an algebraic characterization of regularity properties in bipartite graphs. Our rst approach to regularity in bipartite graphs comes from the study of its spectrum. We characterize these graphs using eigenvalue interlacing and we pro- vide an improved bound for biregular graphs inspired in Guo's inequality. We prove a condition for existence of a k-dominating set in terms of its Laplacian eigenvalues. In particular, we give an upper bound on the sum of the rst Laplacian eigenvalues of a k-dominating set and generalize a Guo's result for these structures. In terms of predistance polynomials, we give a result that can be seen as the biregular coun- terpart of Ho man's Theorem. Finally, we also provide new characterizations of bipartite graphs inspired in the notion of distance-regularity. In Chapter 5 we describe some ideas to work with a result from linear algebra known as the Rayleigh's principle. We observe that the clue is to make the \right choice" of the eigenvector that is used in Rayleigh's principle. We can use this method 1 to give a spectral characterization of regular and biregular partitions. Applying this technique, we also derive an alternative proof for the upper bound of the independence number obtained by Ho man (Chapter 2, Theorem 1.2). Finally, in Chapter 6 other related new results and some open problems are pre- sented.