The scrambling index of primitive digraphs

Abstract

a b s t r a c tIn 2009, Akelbek and Kirkland introduced a useful parameter called the scrambling index of a primitive digraph D, which is the smallest positive integer k such that for every pair of vertices u and v, there is a vertex w such that we can get to w from u and v in D by directed walks of length k. In this paper, we obtain some new upper bounds for the scrambling index of primitive digraphs. Moreover, the maximum index problem, the extremal matrix problem and the index set problem for the scrambling index of various classes of primitive digraphs (e.g. primitive digraphs with d loops, minimally strong digraphs, nearly decomposable digraphs, micro-symmetric digraphs, etc.) are settled, respectively.

Upper Bounds of the Generalized Competition Indices of Symmetric Primitive Digraphs with d Loops

“…The scrambling index of a primitive digraph (D) is denoted as k(D), as introduced in [2,8]. For a primitive digraph (D), we have k(D) = k 1 (D), where k 1 (D) is the 1-competition index of D. The scrambling index of primitive digraphs has been studied by some researchers [9][10][11].…”

Characterization of irreducible Boolean matrices with the largest generalized competition index

“…Note that if m ≥ n, then k m (D) = 1. The m-competition indices of primitive digraphs have been studied by some researchers [1,2,[6][7][8][10][11][12]16]. Kim and Cho [4] have studied the 1-competition index of a strongly connected digraph.…”