Bulletin of the
Korean Mathematical Society BKMS

Cho and Kim \cite{chokim} have introduced the concept of the competition index of a digraph. Similarly, the competition index of an $n \times n$ Boolean matrix $A$ is the smallest positive integer $q$ such that $A^{q+i}(A^T)^{q+i}$ $= A^{q+r+i}(A^T)^{q+r+i}$ for some positive integer $r$ and every nonnegative integer $i$, where $A^T$ denotes the transpose of $A$. In this paper, we study the upper bound of the competition index of a Boolean matrix. Using the concept of Boolean rank, we determine the upper bound of the competition index of a nearly reducible Boolean matrix.

Keywords: competition graph, $m$-step competition graph, competition index, competition period, scrambling index

MSC numbers: 05C20, 05C50