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

TL;DR

This paper investigates the limit of a matrix and graph sequence associated with multipartite tournaments, extending previous work by analyzing cases where the adjacency matrix has no zero rows or sinks.

Contribution

It computes the limit of the sequence \\{C^m(D)\\} for multipartite tournaments with no sinks, generalizing prior results on matrix sequences.

Findings

Sequence \\{A^m(A^T)^m\\} converges if A has no zero rows.

Limit of the graph sequence \\{C^m(D)\\} is determined for tournaments with no sinks.

Provides a graph-theoretical approach to matrix sequence limits.

Abstract

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 $i \neq j$. 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 $\{A^m(A^T)^m\}_{m=1}^{\infty}$. This paper, which is a natural extension of the above paper and was initiated by the observation that $\{A^m(A^T)^m\}_{m=1}^{\infty}$ converges if $A$ has no zero rows, computes the limit of the matrix sequence $\{A^m(A^T)^m\}_{m=1}^{\infty}$ 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 $\left\{ C^m(D) \right\}_{m=1}^{\infty}$ when $D$ has no sinks.