The m-competition indices of symmetric primitive digraphs with loop

For positive integers m and n with 1 � mn, the m-competition index (generalized competition index) of a primitive digraph D of order n is the smallest positive integer k such that for every pair of vertices x and y in D, there exist m distinct vertices v1,v2,...,vm such that there exist walks of length k from x to vi and from y to vi for each i = 1,...,m. In this paper, we study the generalized competition indices of symmetric primitive digraphs without loops. We determine the generalized competition index set and characterize the digraphs in this class with largest generalized competition index.