Biclique covers and partitions of bipartite graphs and digraphs and related matrix ranks of {0,1}-matrice

During the past twenty years there has been considerable research on the biclique cover and partition numbers of bipartite graphs and digraphs and several related matrix ranks. These include the boolean rank, nonnegative integer rank, term rank, and real rank. The main goal of the work in this thesis is to find classes of graphs with equal biclique cover and partition numbers or classes of {0,1}-matrices with at least two of the matrix ranks equal.;In 1991, Lundgren and Maybee showed that all four matrix ranks of nearly reducible matrices are equal. In 1977, Lovasz and Plummer gave bipartite graph representations of nearly decomposable and fully indecomposable matrices. New results in this thesis complement this work. Three of the matrix ranks for nearly decomposable matrices are determined and bipartite graph representations of nearly reducible and irreducible matrices will be given.;In 1991, de Caen proved that the real rank of an n-tournament matrix is at least n -- 1. This lower bound implies that the term rank of an n-tournament matrix is also at least n -- 1. A new result in this thesis characterizes those tournaments with term rank n. The classification of singular tournaments remains an open problem, but some results are known for specific subclasses of tournaments. In 1990, Shader characterized singular upset tournament matrices and proved that the nonnegative integer rank of upset tournament matrices is equal to the real rank.;New results concerning the boolean rank and term rank of upset tournament matrices are discussed in this thesis, Specifically, a characterization of upset tournament matrices with respect to their boolean rank and a best possible lower bound for the boolean rank is given. In addition, it is shown that the number of nonisomorphic upset tournaments with equal biclique cover and partition numbers can be given in terms of convolutions of the Fibonacci sequence. These results, together with Shader's work, give a complete characterization of upset tournament matrices with respect to each rank and with respect to their biclique cover and partition numbers.