The m-competition indices of symmetric primitive digraphs without loops

Abstract

Abstract. For positive integers m and n with 1 ≤ m ≤ n, 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 v 1 , v 2 , . . . , vm such that there exist walks of length k from x to v i and from y to v i 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.

The kth Local Exponent of Doubly Symmetric Primitive Digraphs with d Loops

“…It is important to mention that the kth local exponent of such a class of digraphs has not been studied before. Using graph theory methods, we obtain the upper bound of the kth local exponent of digraphs in DS n (d), where 1 ≤ k ≤ n. Some studies have investigated the scrambling index [13][14][15][16] and generalized competition index [17][18][19][20][21][22][23]. Several studies explored the generalized µ-scrambling indices, please refer to [24][25][26].…”

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

“…There has been interest recently in a generalized competition index [12][13][14][15][16][17]. Suppose a memoryless communication system is represented by a primitive digraph of n vertices; then, the m-competition indices represent the longest first time for m vertices to know all 2 bits of the information (see [5,18]).…”

The Generalized Competition Indices of Doubly Symmetric Primitive Digraphs with d Loops

“…There exist many studies about exponents and their generalization; for example [9][10][11]. For some studies on generalized competition indices, see [12][13][14][15][16].…”