A bound of generalized competition index of a primitive digraph

Kim, Hwa Kyung; Park, Sung Gi

doi:10.1016/j.laa.2011.06.040

Cited by 12 publications

(6 citation statements)

References 6 publications

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“…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].…”

Section: We Easily Get γmentioning

confidence: 99%

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

Chen

2022

Symmetry

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Let D be a primitive digraph of order n. The exponent of a vertex x in V(D) is denoted γD(x), which is the smallest integer q such that for any vertex y, there is a walk of length q from x to y. Let V(D)={v1,v2,⋯,vn}. We order the vertices of V(D) so that γD(v1)≤γD(v2)≤⋯≤γD(vn) is satisfied. Then, for any integer k satisfying 1≤k≤n,γD(vk) is called the kth local exponent of D and is denoted by expD(k). Let DSn(d) represent the set of all doubly symmetric primitive digraphs with n vertices and d loops, where d is an integer such that 1≤d≤n. In this paper, we determine the upper bound for the kth local exponent of DSn(d), where 1≤k≤n. In addition, we find that the upper bound for the kth local exponent of DSn(d) can be reached, where 1≤k≤n.

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Section: We Easily Get γmentioning

confidence: 99%

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

Chen

2022

Symmetry

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“…There has been interest recently in generalized competition index [5,6,7,9]. Let S 0 n denote the set of all symmetric primitive digraphs of order n without loops.…”

Section: Then We Havementioning

confidence: 99%

The m-competition indices of symmetric primitive digraphs without loops

Shao¹,

Gao²,

Li³

2012

ELA

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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.

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“…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]).…”

Section: Introductionmentioning

confidence: 99%

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

Chen

2023

Symmetry

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A digraph (D) is symmetric if (u,v) is an arc of D and if (v,u) is also an arc of D. If a symmetric digraph is primitive and contains d loops, then it is said to be a symmetric primitive digraph with d loops. The m-competition index (generalized competition index) of a digraph is an extension of the exponent and the scrambling index. The m-competition index has been applied to memoryless communication systems in recent years. In this article, we assume that Sn(d) represents the set of all symmetric primitive digraphs of n vertices with d loops, where 1≤d≤n. We study the m-competition indices of Sn(d) and give their upper bounds, where 1≤m≤n. Furthermore, for any integer m satisfying 1≤m≤n, we find that the upper bounds of the m-competition indices of Sn(d) can be reached.

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A bound of generalized competition index of a primitive digraph

Cited by 12 publications

References 6 publications

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

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

The m-competition indices of symmetric primitive digraphs without loops

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

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