New approach to the k-independence number of a graph

Let G = (V,E) be a graph and k > 0 an integer. A k-independent set S V is a set of vertices such that the maximum degree in the graph induced by S is at most k. With k(G) we denote the maximum cardinality of a k-independent set of G. We prove that, for a graph G on n vertices and average degree d, k(G) > k+1 dde+k+1n, improving the hitherto best general lower bound due to Caro and Tuza [Improved lower bounds on k-independence, J. Graph Theory 15 (1991), 99–107].Peer ReviewedPostprint (published version

Binary matrix factorisations under Boolean arithmetic

For a binary matrix X, the Boolean rank br(X) is the smallest integer for which X can be factorised into the Boolean matrix product of two binary matrices A and B with inner dimension br(X). The isolation number i(X) of X is the maximum number of 1s no two of which are in a same row, column or a 2 x 2 submatrix of all 1s. In Part I. of this thesis, we continue Anna Lubiw's study of firm matrices. X is said to be firm if i(X)=br(X) and this equality holds for all its submatrices. We show that the stronger concept of superfirmness of X is equivalent to having no odd holes in the rectangle cover graph of X, the graph in which br(X) and i(X) translate to the clique cover number and the independence number, respectively. A binary matrix is minimally non-firm if it is not firm but all of its proper submatrices are. We introduce a matrix operation that leads to generalised binary matrices and, under some conditions, preserves firmness and superfirmness. Then we use this matrix operation to derive several infinite families of minimally non-firm matrices. To the best of our knowledge, minimally non-firm matrices have not been studied before and our constructions provide the first infinite families of them. In Part II. of this thesis, we explore rank-k binary matrix factorisation (k-BMF). In k-BMF, we are given an m x n binary matrix X with possibly missing entries and need to find two binary matrices A and B of dimension m x k and k x n respectively, which minimise the distance between X and the Boolean matrix product of A and B in the squared Frobenius norm. We present a compact and two exponential size integer programs (IPs) for k-BMF and show that the compact IP has a weak LP relaxation, while the exponential size IPs have a stronger equivalent LP relaxation. We introduce a new objective function, which differs from the traditional squared Frobenius objective in attributing a weight to zero entries of the input matrix that is proportional to the number of times a zero is erroneously covered in a rank-k factorisation. For one of the exponential size IPs we describe a computational approach based on column generation. Experimental results on synthetic and real word datasets suggest that our integer programming approach is competitive against available methods for k-BMF and provides accurate low-error factorisations

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

Application of new probabilistic graphical models in the genetic regulatory networks studies

This paper introduces two new probabilistic graphical models for reconstruction of genetic regulatory networks using DNA microarray data. One is an Independence Graph (IG) model with either a forward or a backward search algorithm and the other one is a Gaussian Network (GN) model with a novel greedy search method. The performances of both models were evaluated on four MAPK pathways in yeast and three simulated data sets. Generally, an IG model provides a sparse graph but a GN model produces a dense graph where more information about gene-gene interactions is preserved. Additionally, we found two key limitations in the prediction of genetic regulatory networks using DNA microarray data, the first is the sufficiency of sample size and the second is the complexity of network structures may not be captured without additional data at the protein level. Those limitations are present in all prediction methods which used only DNA microarray data.Comment: 38 pages, 3 figure

Reasoning about Independence in Probabilistic Models of Relational Data

We extend the theory of d-separation to cases in which data instances are not independent and identically distributed. We show that applying the rules of d-separation directly to the structure of probabilistic models of relational data inaccurately infers conditional independence. We introduce relational d-separation, a theory for deriving conditional independence facts from relational models. We provide a new representation, the abstract ground graph, that enables a sound, complete, and computationally efficient method for answering d-separation queries about relational models, and we present empirical results that demonstrate effectiveness.Comment: 61 pages, substantial revisions to formalisms, theory, and related wor

Hierarchical Models for Independence Structures of Networks

We introduce a new family of network models, called hierarchical network models, that allow us to represent in an explicit manner the stochastic dependence among the dyads (random ties) of the network. In particular, each member of this family can be associated with a graphical model defining conditional independence clauses among the dyads of the network, called the dependency graph. Every network model with dyadic independence assumption can be generalized to construct members of this new family. Using this new framework, we generalize the Erd\"os-R\'enyi and beta-models to create hierarchical Erd\"os-R\'enyi and beta-models. We describe various methods for parameter estimation as well as simulation studies for models with sparse dependency graphs.Comment: 19 pages, 7 figure