A bound on the generalized competition index of a primitive matrix using Boolean rank

For a positive integer m where 1⩽m⩽n, the m-competition index (generalized competition index) of a primitive digraph is the smallest positive integer k such that for every pair of vertices x and y, there exist m distinct vertices v1,v2,…,vm such that there are directed walks of length k from x to vi and from y to vi for 1⩽i⩽m. The m-competition index is a generalization of the scrambling index and the exponent of a primitive digraph. In this study, we determine an upper bound on the m-competition index of a primitive digraph using Boolean rank and give examples of primitive Boolean matrices that attain the bound.