  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 $A$.
