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

Abstract

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 v 1 , v 2 , … , v m such that there are directed walks of length k from x to v i and from y to v i 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.