The <i>k</i>‐degree Cayley graph and its topological properties

Hsieh, Sun-Yuan; Hsiao, Tien-Te

doi:10.1002/net.20096

Cited by 13 publications

(11 citation statements)

References 27 publications

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“…As a consequence of this result, some topological properties such as maximal fault tolerance and regularity of degree k shown in [1] are seen to be valid.…”

Section: Resultsmentioning

confidence: 72%

“…Nevertheless, the proof in [1] depends on a particular permutation group and its generators implicitly given in Appendix A.…”

Section: Form We Sometimes Use V M To Represent a Node V With The Mamentioning

confidence: 99%

“…[1]. A k-degree Cayley graph G k,n is an undirected graph with N = n(k − 1) n nodes for any integers n ≥ 2 and k ≥ 3.…”

Section: Introductionmentioning

confidence: 99%

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A note on the k‐degree Cayley graph

Tanaka

Shibata

2009

Networks

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The k -degree Cayley graph has been proposed by Hsieh and Hsiao (Networks 47 (2006), 26-36). They have shown that the k -degree Cayley graph is actually a Cayley graph. According to their definition, a generating set of that graph was defined implicitly. Therefore, it requires some effort to figure out that such graphs are actually Cayley graphs. In this article, we use the wreath product of groups to show that k -degree Cayley graphs form a class of Cayley graphs.

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“…As a consequence of this result, some topological properties such as maximal fault tolerance and regularity of degree k shown in [1] are seen to be valid.…”

Section: Resultsmentioning

confidence: 72%

“…Nevertheless, the proof in [1] depends on a particular permutation group and its generators implicitly given in Appendix A.…”

Section: Form We Sometimes Use V M To Represent a Node V With The Mamentioning

confidence: 99%

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A note on the k‐degree Cayley graph

Tanaka

Shibata

2009

Networks

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“…For example, a number of networks of both theoretical and practical importance, including hypercubes, alternating group graphs, star graphs and some of their generalisations are both vertextransitive and edge-transitive. The reader is referred to the papers [15], [16], [21], [49] for some results pertaining to graphs with high symmetric properties as models for interconnection networks.…”

Section: Introductionmentioning

confidence: 99%

Symmetric Property and Reliability of Balanced Hypercube

Zhou

Yang

et al. 2015

IEEE Trans. Comput.

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Huang and Wu in [IEEE Transactions on Computers 46 (1997) 484-490] introduced the balanced hypercube BH n as an interconnection network topology for computing systems, and they proved that BH n is vertex-transitive. However, some other symmetric properties, say edge-transitivity and arctransitivity, of BH n remained unknown. In this paper, we solve this problem and prove that BH n is an arc-transitive Cayley graph. Using this, we also investigate some reliability measures, including super-connectivity, cyclic connectivity, etc., in BH n . First, we prove that every minimum edge-cut of BH n ðn ! 2Þ isolates a vertex, and every minimum vertex-cut of BH n ðn ! 3Þ isolates a vertex. This is stronger than that obtained by Wu and Huang which shows the connectivity and edge-connectivity of BH n are 2n. Second, Yang [Applied Mathematics and Computation 219 (2012) 970-975.] proved that for n ! 2, the super-connectivity of BH n is 4n À 4 and the super edge-connectivity of BH n is 4n À 2. In this paper, we proved that BH n ðn ! 2Þ is super-0 but not super-k 0 . That is, every minimum super edge-cut of BH n ðn ! 2Þ isolates an edge, but the minimum super vertex-cut of BH n ðn ! 2Þ does not isolate an edge. Third, we also obtain that for n ! 2, the cyclic connectivity of BH n is 4n À 4 and the cyclic edge-connectivity of BH n is 4ð2n À 2Þ. That is, to become a disconnected graph which has at least two components containing cycles, we need to remove at least 4n À 4 vertices (resp. 4ð4n À 2Þ edges) from BH n ðn ! 2Þ.

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“…New classes of interconnection networks with odd fix degrees are also developed in [7,22]. Hsieh and Hsiao develop a new family Cayley graph G k,n with k-degree, which possess a useful property in that the degree of each vertex is fixed by a general positive integer k without regard to the number of vertices [12].…”

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confidence: 99%

A new family of Cayley graph interconnection networks based on wreath product and its topological properties

Zhang

Xiao

2011

Cluster Comput

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This paper introduces a new type of Cayley graphs for building large-scale interconnection networks, namely WG 2m n , whose vertex degree is m + 3 when n ≥ 3 and is m + 2 when n = 2. A routing algorithm for the proposed graph is also presented, and the upper bound of the diameter is deduced as 5n/2 . Moreover, the embedding properties and maximal fault tolerance are also analyzed. Finally, we compare the proposed networks with some other similar network topologies. It is found that WG 2m n is superior to other interconnection networks because it helps to construct large-scale networks with lower cost.

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The k‐degree Cayley graph and its topological properties

Cited by 13 publications

References 27 publications

A note on the k‐degree Cayley graph

A note on the k‐degree Cayley graph

Symmetric Property and Reliability of Balanced Hypercube

A new family of Cayley graph interconnection networks based on wreath product and its topological properties

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