THE COMPETITION INDEX OF A NEARLY REDUCIBLE BOOLEAN MATRIX

Abstract

Cho and Kim [4] have introduced the concept of the competition index of a digraph. Similarly, the competition index of an <TEX>$n{\times}n$</TEX> Boolean matrix A is the smallest positive integer q such that <TEX>$A^{q+i}(A^T)^{q+i}=A^{q+r+i}(A^T)^{q+r+i}$</TEX> for some positive integer r and every nonnegative integer i, where <TEX>$A^T$</TEX> 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.