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

Given a positive integer $m$, the $m$-step competition graph of a digraph $D$, denoted by $C^m(D)$, has the same vertex set as $D$ and has an edge between vertices $u$ and $v$ if and only if there exists a vertex $w$ such that there exist directed walks of length $m$ from $u$ to $w$ and from $v$ to $w$, respectively. In this paper, we completely characterize the convergence of $\{C^m(D)\}_{m=1}^{\infty}$ for a multipartite tournament $D$ based on the last nontrivial strong component of $D$. Furthermore, not only do we determine the limit in the case of convergence, but also in the event of divergence, we specify how $C^m(D)$ changes periodically depending on the value of $m$. Our results extend the work of Jung et al. [On the limit of the sequence $\{C^m (D)\}_{m=1}^{\infty}$ for a multipartite tournament $D$. Discrete Appl. Math., 340:1--13, 2023] which addresses the case of the last strong component being nontrivial, thereby completing the convergence analysis of $\{C^m(D)\}_{m=1}^{\infty}$ for a multipartite tournament $D$. Our results can also be expressed in terms of matrix sequence $\{A^m(A^T)^m\}_{m=1}^{\infty}$ for the adjacency matrix $A$ of $D$ and this part is also covered in the text.