Generalized competition index of an irreducible Boolean matrix

For a positive integer m, where 1⩽m⩽n, the m-competition graph of an irreducible Boolean matrix A of order n, denoted by Cm(A), is the graph that has the same vertex set as its digraph D(A), and there is an edge between vertices x and y (x≠y) if and only if there exist m distinct vertices v1,v2,…,vm such that x→vi and y→vi for 1⩽i⩽m in D(A). The smallest positive integer q such that Cm(Aq+i)=Cm(Aq+r+i) for some positive integer r and every nonnegative integer i is called the m-competition index (generalized competition index) of A. The m-competition index is a generalization of the competition index and the index of an irreducible Boolean matrix. In this study, we determine the upper bound on the m-competition index of an irreducible Boolean matrix.