Competition Indices of Strongly Connected Digraphs

Abstract

Abstract. Cho and Kim [4] and Kim [6] introduced the concept of the competition index of a digraph. Cho and Kim [4] and Akelbek and Kirkland [1] also studied the upper bound of competition indices of primitive digraphs. In this paper, we study the upper bound of competition indices of strongly connected digraphs. We also study the relation between competition index and ordinary index for a symmetric strongly connected digraph.

The scrambling index of primitive digraphs

“…Since exp(D, 1) ≤ 3 [10,2], and D([A(D)] ) ∈ T n for D ∈ T n (n > 6), the result in [4] can be obtained immediately:…”

“…For example, let S n denote the set of all primitive symmetric digraphs of order n [2], and let T n denote the set of all primitive tournaments of order n [4,1]. It is known that exp(D, 1) ≤ n − 1 for D ∈ S n [2].…”

“…Observe that the scrambling index is the competition index [4] in the case of primitive digraphs. In [4], Cho and Kim gave an upper bound on the competition index of a primitive digraph D of order n and girth s. In [3,5], Akelbek and Kirkland improved the upper bound given by Cho and Kim, and then they settled the MIP and EMP for the scrambling index of primitive digraphs of order n. In [6], Kim investigated the competition index of tournaments. In [7], the MIP, EMP and ISP for the scrambling index of primitive symmetric digraphs are studied, respectively.…”

The Competition Index of a Nearly Reducible Boolean Matrix

“…Cho and Kim [4] presented the following result regarding the upper bound of the competition index of a strongly connected digraph. Proposition 1.2 (Cho and Kim [4]).…”

“…Cho and Kim [4] have introduced the concept of the competition index of a digraph. Similarly, the competition index of an n × n Boolean matrix A is the smallest positive integer q such that A q+i (A T ) q+i = A q+r+i (A T ) q+r+i for some positive integer r and every nonnegative integer i, where A T denotes the transpose of A.…”