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In this paper, we extend the results given by Park {\em et al.} \cite{ppk} bystudying 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 connectedwith an arbitrary number of strong components. In the process forgeneralization, we concretize ideas behind their arguments. We completelycharacterize $A$ for which $\{\Gamma(A^m)\}_{m=1}^\infty$ converges. Then wefind its limit when all of the irreducible diagonal blocks are of order atleast 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 theseresults are derived by studying the $m$-step competition graph of the digraphof $A$.

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