Characterization of irreducible Boolean matrices with the largest generalized competition index

For a positive integer m, 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. 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. In this paper, we characterize irreducible Boolean matrices with the largest generalized competition index.