On the limit of the sequence ${\{C^m(D)\}}_{m=1}^{\mathrm{\infty }}$ for a multipartite tournament $D$

For an integer $k\ge 2$, let $A$ be a Boolean block matrix with blocks $A_{ij}$ for $1\le i,j\le k$ such that $A_{ii}$ is a zero matrix and $A_{ij}+A_{ji}^T$ is a matrix with all elements $1$ but not both corresponding elements of $A_{ij}$ and $A_{ji}^T$ equal to $1$ for $i\mathrm{\ne }j$. Jung~{\em et al.} [Competition periods of multipartite tournaments. {\it Linear and Multilinear Algebra}, https://doi.org/10.1080/03081087.2022.2038057] studied the matrix sequence $\{A^m(A^T{)}^m{\}}_{m=1}^{\mathrm{\infty }}$. This paper, which is a natural extension of the above paper and was initiated by the observation that $\{A^m(A^T{)}^m{\}}_{m=1}^{\mathrm{\infty }}$ converges if $A$ has no zero rows, computes the limit of the matrix sequence $\{A^m(A^T{)}^m{\}}_{m=1}^{\mathrm{\infty }}$ if $A$ has no zero rows. To this end, we take a graph theoretical approach: noting that $A$ is the adjacency matrix of a multipartite tournament $D$, we compute the limit of the graph sequence ${\{C^m(D)\}}_{m=1}^{\mathrm{\infty }}$ when $D$ has no sinks.