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

Abstract

Let DSn(d) denote the set of all doubly symmetric primitive digraphs of order n with d loops, where d is an integer and 1≤d≤n. In this paper, we determine the upper bounds for the m-competition indices(generalized competition indices) of DSn(d), where 1≤m≤n. If n and d satisfy that n is odd and d is odd, or n≤2d−2 and d is even such that d≥2, then the upper bounds for the m-competition indices of DSn(d) can be reached, where 1≤m≤n.

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].…”

“…If D ∈ DS n (d), then there are two connected subgraphs D * and D * * of D. In addition, there is a unique path for any two different vertices in D * and D * * , respectively. Moreover, there is a unique path for any two different vertices in D. Let x, y be any pair of vertices of D such that x ∈ V * and y ∈ V * * , then V(P(x, v n+1 [23]). After removing d loops from D, the obtained graph is a tree.…”

“…In Lemma 2 in [23], let D ∈ DS n (d), where n is even and d be even such that d ≥ 2. We can directly get the following Lemma 6.…”

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]).…”