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

TL;DR

This paper investigates the convergence behavior of matrix sequences derived from Boolean matrices with linearly connected digraphs, providing characterizations of convergence, limits, and specific structural conditions.

Contribution

It generalizes previous results by characterizing convergence and limits of the matrix sequence for more complex digraph structures with multiple strong components.

Findings

Characterized matrices for which the sequence converges.

Determined the limit when all diagonal blocks are at least size two.

Identified conditions for the limit to be a block diagonal matrix.

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 $A$.