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

Abstract

In this paper, we extend the results given by Park {\em et al.} \cite{ppk} by studying the convergence of the matrix sequence $\{\Gamma (A^m){\}}_{m=1}^{\infty }$ for a matrix $A\in {\mathcal{B}}_n$ 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 $\{\Gamma (A^m){\}}_{m=1}^{\infty }$ 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 $\{\Gamma (A^m){\}}_{m=1}^{\infty }$ is a $J$ block diagonal matrix. All of these results are derived by studying the $m$-step competition graph of the digraph of $A$.

19 pages, 4 figures