Abstract: Let D be a primitive digraph of order n.For positive integers m and n(1≤m≤n), we define its mcompetition index,km(D), as the smallest positive integer k that satisfies the condition that for every pair of vertices x and y, there are m different vertices v1,v2,…，vm in D that satisfy xkvi and ykvi(i=1,2,…，m). We address a primitive digraph D of order n with two (n-2) cycles and one (n-3) cycle. We derive primitive digraphs Dn-2 and Dn-3 based on the structure of D. We can further obtain m-competition index of the primitive digraph D based on the definition of m-competition index.

Key words: digraph, primitive digraph, m-competition index