On the matrix sequence {Gamma(A^m)}_{m=1}^infinity for a Boolean matrix A whose digraph is linearly connected

Abstract

In this thesis, we extend the results given by Park et al. [12] by studying the convergence of the matrix sequence {Gamma(A^m)}_{m=1}^infinity for a matrix A in {B}_n the digraph of which is linearly connected with an arbitrary number of strong components. In the process for generalization, we concretize ideas behind their arguments. We completely characterize A for which {Gamma(A^m)}_{m=1}^infinity converges. Then we find its limit when all of the irreducible diagonal blocks are of order at least two. We go further to characterize A for which the limit of {Gamma(A^m)}_{m=1}^infinity is a J block diagonal matrix. All of these results are derived by studying the m-step competition graph of the digraph of A.