Generalized competition index of a primitive digraph

Abstract

For positive integers k and m , and a digraph D , the k -step m -competition graph C m k ( D ) of D has the same set of vertices as D and an edge between vertices x and y if and only if there are distinct m vertices v 1 , v 2 , … , v m in D such that there are directed walks of length k from x to v i and from y to v i for 1 ⩽ i ⩽ m . In this paper, we present the definition of m -competition index for a primitive digraph. The m -competition index of a primitive digraph D is the smallest positive integer k such that C m k ( D ) is a complete graph. We study m -competition indices of primitive digraphs and provide an upper bound for the m -competition index of a primitive digraph.