On the matrix sequence {Γ(Am)}m=1∞ for a Boolean matrix A whose digraph is linearly connected

In this paper, we extend the results given by Park et al. [12] by studying the convergence of the matrix sequence {Γ(Am)}m=1∞ for a matrix A∈Bn 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 {Γ(Am)}m=1∞ 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 {Γ(Am)}m=1∞ is a J block diagonal matrix. All of these results are derived by studying the m-step competition graph of the digraph of A.