A bound of generalized competition index of a primitive digraph

For a positive integer m, where 1≤m≤n, the m-competition index (generalized competition index) of a primitive digraph D 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 exist 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 paper, we study the upper bound of the m-competition index of a primitive digraph using its order and girth.