Nearly reducible and nearly decomposable - special classes of irreducible and fully indecomposable matrices

The purpose of this dissertation is to examine the structural properties of matrices whose entries are either 0 or 1. There are three main results. In Theorem 1, the author shows that the maximal number of positive entries (arcs) in an n x n nearly reducible matrix (minimally connected graph with n vertices) is 2(n - 1) and the matrix has a canonical form. In Theorem 2, he argues that the maximal number of positive entries in a nearly decomposable n x n matrix is 3(n - 1) and is obtained uniquely at a canonical matrix. In Theorem 3, he examines the structure of those nearly decomposable (0,l)-matrices whose permanent equals sigma - 2n + 2 where sigma is the number of positive entries in A.