2-Restricted Edge Connectivity of Wheel Networks

Feng, Wei; Wang, Shiying

doi:https://doi.org/10.1155/2021/3725778

Mathematical Problems in Engineering

On this page

Abstract Introduction Preliminaries Conclusion Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2021 | Article ID 3725778 | https://doi.org/10.1155/2021/3725778

2-Restricted Edge Connectivity of Wheel Networks

Wei Feng^1,2and Shiying Wang¹

Academic Editor: Qiuhong Wu

Received26 Apr 2021

Accepted26 Jun 2021

Published06 Jul 2021

Abstract

The-restricted edge connectivity (-REC) is an efficient index in evaluating the reliability and fault tolerance of large-scale processing systems. Assume that is a simple connected graph and . The -REC of , denoted by , is the minimum such that is disconnected and any component has at least nodes. The -dimensional wheel network , a kind of Cayley graph, possesses many desired features. In this paper, we establish that is for , and is -optimal.

1. Introduction

With the increase of interconnection networks’ scale, the probability of processors and/or links’ failing increases. How to evaluate the reliability and fault tolerance of a large-scale processing system has become a significant research topic. A graph can be always used to represent a practical interconnection network, where (resp. ) represents the set of all processors (resp. links). One of the classical measurements for evaluating the reliability and fault tolerance of a network is edge connectivity of a graph . Generally, the larger the is, the more reliable the responding network is. However, this index has its shortcoming, that is, it tacitly supposes that each edge incidents with the same node in may fail simultaneously. It is very unlikely in the process of applications of large-scale networks.

One of methods for compensating this deficiency is to increase certain conditions in each component of in the concept of . The -restricted edge connectivity (-REC) was introduced by Fàbrega and Fiol [1]. Assume that is a simple connected graph and . The -REC of , denoted by , is the minimum such that is disconnected and any component has at least nodes. Clearly, if , then . In particular, is equal to the conditional edge connectivity introduced in [2]. In [3], it is shown that the larger is, the more reliable the corresponding network is. Esfahanian et al. [2] have established that can be found by solving no more than network flow problems and if exists, where . A graph with is called a -optimal graph. 2-REC and -optimal graph have been discussed by many researchers [4–8].

2. Preliminaries

2.1. Terminology and Notations

For those terminology and notations not stated here, the readers can refer to [9, 10]. Suppose that is an undirected and simple graph. For any node , define (resp. is incident with in ) to be the neighborhood (resp. edge neighborhood) of in . The degree of a node is . (resp. ) is the minimum (resp. maximum) degree of a graph . For any , let . The length of a shortest cycle of is defined as the girth of . For , let represent the induced subgraph of in . For , let and . If , for any , then the graph is -regular. The subscript of , and are usually omitted if is already known from the context. A graph is node transitive if there exist such that , for , where is the automorphism group of . Considering two graphs and , means that they are isomorphic.

2.2. The Wheel Networks

Assume that is a finite group. Suppose that a spanning set of satisfies , where is the identity element of . A directed Cayley graph can be stated as follows: its node set is , and its directed arc set is . If holds for , then is an undirected graph.

Assume that is a connected and simple graph and . If we consider each edge in as a transposition of the symmetric graph , then the edge set of can correspond exactly to a transposition set in . From this sense, is usually called a transposition simple graph. The Cayley graph Cay is called the corresponding Cayley graph of , also denoted by Cay . By Akers et al. [11], . If is a tree (resp. path, star), then the corresponding Cayley graph, denoted by (resp. , ) is called a transposition tree (resp. bubble-sort graph, star graph) [11]. If , where and , then the corresponding Cayley graph , denoted by , is called a bubble-sort star graph [12]. If , where and , then the corresponding Cayley graph , denoted by , is called a -dimensional wheel network [13]. In other words, = . And, for any , iff , , or , , or . The diameters of and are both , where [12, 14]. can be embedded in a . surpasses in connectivity [15, 16]. Hence, is also a desirable processing model. is studied by many researchers [13, 14, 17–25]. for has been depicted in Figure 1.

The permutation can be conveniently denoted by . Let and be two permutations. The product means the composition function followed by , for example, .

If we delete the node labeling in the transposition simple graph of , then a graph on node set is obtained. Note that the Cayley graph responding is . Hence, can be decomposed into copies of as follows. For any , suppose that is the subgraph of induced by the following node set ranges over any permutations of . Hence, . For any , then three neighbors , , and of are all called outside neighbors of , denoted by , , and , respectively. An edge is called a cross-edge in the given decomposition if are in distinct ’s.

Proposition 1. (see [9]). .

Lemma 1. (see [15]). , .

Proposition 2. (see [12]). is -regular, .

Proposition 3. (see [26]). , is -regular, node transitive.

Proposition 4. (see [26]). , is bipartite.

Proposition 5. (see [16]). , where . Then, , , and belong to 3 distinct ’s .

Proposition 6. (see [16]). The number of independent cross-edges between two distinct ’s is .

Proposition 7. (see [27]). For , where , then .

Lemma 2. (see [2]). .

3. 2-REC of -Dimensional Wheel Network

Theorem 1. , for , and is a -optimal graph.

Proof. By Propositions 3 and 4, we can obtain . By Lemma 2, , for . Next, we prove , for .
We decompose into disjoint copies as above, where each . Assume that is any edge subset of with such that has no isolated nodes. We just need to prove that is connected. Assume that , , and . Clearly, . We partition into two parts: and , where and . Obviously, . Let . We distinguish two cases for .

Case 1. .

Claim 1. If , then is connected.

Proof. Note that corresponding to in satisfies that . By Propositions 1 and 2 and Lemma 1, , for . Hence, every is connected, for any in . By Proposition 6 and for , hence, is connected to for any two distinct and in . We deduce that is connected. Claim 1 is completed.
Since , for , we have that . We prove that is connected in the following situations.

Case 1.1. .
Using a proof similar to that in Claim 1, we can yield that is connected.

Case 1.2. , .
Let be the only one copy satisfying . Let be any component of . Next, we prove that is connected to . Note that has no isolated nodes. If contains only a node , then is connected to by one of , , and . If , then has an edge . Let and . By Propositions 5 and 7, there is a path in such that . So, is connected to by . We deduce that is connected.

Case 1.3. and .
Let be just the two copies satisfying . It is easy to see that and . Note that for , where corresponding to belongs to . We have that and are connected. Similarly, we can deduce that and are connected. By Claim 1, is connected.

Case 2. .
Since for , we have that . We prove that is connected in the following cases:

Case 2.1. and .
Using a proof similar to that in Claim 1, we can deduce that is connected for . We say that is connected. Otherwise, there are and , and they are disconnected, where and are both in . By Proposition 6, . Note that . Hence, there is a copy in such that and , where . We obtain that and are both connected to . Therefore, we have that is connected.

Case 2.2. and .
Let be the only one copy satisfying . Using a proof similar to that in Claim 1, we can include that is connected for each in . We say that is connected. Otherwise, there are and in , and they are disconnected in . By Proposition 6, we have that . Note that holds clearly. Hence, there is a copy in such that and , where . We obtain that and are both connected to . Therefore, we have that is connected.
Let be any component of . Next, we prove that is connected to . If contains only a node , then is connected to by one of , , and , since has no isolated nodes. If , then has an edge . Let and . By Propositions 5 and 7, there is a path in such that , where . So, is connected to by . We deduce that is connected.

Case 2.3. and .
Let be just the two copies satisfying . Using a proof similar to that in Claim 1, we include that is connected for each in . Clearly, , where . By Proposition 6, is connected.

Case 2.3.1. Both and are connected.
We claim that is connected. Otherwise, there are two copies and are disconnected for . By Proposition 6, . Note that . Hence, for any , such that both and are connected to . We deduce that is connected.

Case 2.3.2. Only one of and is connected.
Assume that be disconnected and be connected. By Proposition 6 and , is connected to , where is any one in . Therefore, is connected to . Let be any component of . Next, we prove that is connected to . If contains only a node , then is connected to by one of , , and , since has no isolated nodes. If , then has an edge . Let and . By Propositions 5 and 7, there is a path in such that , where . So, is connected to by . We deduce that is connected.

Case 2.3.3. Neither nor is connected.
Let be all components in , and let be all components in . We consider any for . Let , and we give Claim 2 as follows:

Claim 2. is connected to .

Proof. By Proposition 5, it is easy to prove that, at most one cross-edge of is in , where . We distinguish two cases to complete Claim 2.

Case 2.3.3.1. .
If , then it is a contradiction to that has no isolated nodes. Hence, there is an element and . By Proposition 5, is connected to by . Claim 2 is completed in this case.

Case 2.3.3.2. .
Suppose that . Since and , we can deduce . Note that by Proposition 5. So, there is an element and such that is connected to by . The situations of and can be similarly proved. Claim 2 is completed in this case.
Similarly, we can deduce that a node in any for is connected to . By the arbitrariness of and , we prove that each of is connected to . So, is connected.

Case 2.4. and .
Let be just the three copies satisfying . It implies that since . Hence, there is no edges of in . It implies that , for , and , for . Note that . Hence, and are connected. Similarly, both and are connected to . We have that is connected.
Therefore, we established for . Since , we say that is a -optimal graph. The theorem is completed.

4. Conclusion

The -restricted edge connectivity is a generalization of classical edge connectivity. It is a more refined index in measuring fault tolerance of interconnection networks. Here, we obtain that 2-restricted edge connectivity of wheel network . Note that wheel network is a kind of regular graph. A nontrivial graph with order whose degree set contains elements is named an antiregular graph [28]. It would be interesting to find the -restricted edge connectivity of antiregular graphs.

Data Availability

The data used to support the findings of the study are available from the corresponding author upon request.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work was supported by the NSFC (61772010), the Higher Educational Scientific Research Projects of Inner Mongolia (NJZY21439), the project of IMUN (NMDGP17106), and the 2020 Scientific Research Project for Postgraduates of Henan Normal University (YL202008).

References

J. Fàbrega and M. A. Fiol, “Extraconnectivity of graphs with large girth,” Discrete Mathematics, vol. 127, pp. 163–170, 1994.
View at: Google Scholar
A.-H. Esfahanian and S. L. Hakimi, “On computing a conditional edge connectivity of a graph,” Information Processing Letters, vol. 27, no. 4, pp. 195–199, 1988.
View at: Publisher Site | Google Scholar
M. Wang and Q. Li, “Conditional edge connectivity properties, reliability comparisons and transitivity of graphs,” Discrete Mathematics, vol. 258, no. 1–3, pp. 205–214, 2002.
View at: Publisher Site | Google Scholar
Q. Li and Q. Li, “Reliability analysis of circulants,” Networks, vol. 31, no. 2, pp. 61–65, 1998.
View at: Publisher Site | Google Scholar
Q. L. Li and Q. Li, “Super edge connectivity properties of connected edge symmetric graphs,” Networks, vol. 33, pp. 147–159, 1999.
View at: Publisher Site | Google Scholar
J. Meng, “Optimally super-edge-connected transitive graphs,” Discrete Mathematics, vol. 260, no. 1–3, pp. 239–248, 2003.
View at: Publisher Site | Google Scholar
J.-M. Xu and K.-L. Xu, “On restricted edge-connectivity of graphs,” Discrete Mathematics, vol. 243, no. 1–3, pp. 291–298, 2002.
View at: Publisher Site | Google Scholar
M. Xu and J. Jing, “The connectivity and super connectivity of bubble-sort graph,” Acta Mathematicae Applicatae Sinica, vol. 35, no. 5, pp. 789–794, 2012, in Chinese.
View at: Google Scholar
J. A. Bondy and U. S. R. Murty, Graph Theory, Springer, New York, NY, USA, 2007.
W. Thomas, Hungerford, Algebra, Springer, New York, NY, USA, 1974.
S. B. Akers and B. Krishnamurthy, “A group-theoretic model for symmetric interconnection networks,” IEEE Transactions on Computers, vol. 38, no. 4, pp. 555–566, 1989.
View at: Publisher Site | Google Scholar
Z.-T. Chou, C.-C. Hsu, and J.-P. Sheu, “Bubble-sort star graphs: a new interconnection network,” in Proceedings of the International Conference on Parallel and Distributed Systems, pp. 41–48, Tokyo, Japan, June 1996.
View at: Google Scholar
H. Shi and J. Lu, “On conjectures of interconnection networks,” Computer Engineering and Applications, vol. 44, no. 31, pp. 112–115, 2008, in Chinese.
View at: Google Scholar
H. Shi, F. Hou, J. Ma, and G. Wang, “Study on diameter and average distance of wheel network,” Journal of Gansu Sience, vol. 24, no. 2, pp. 103–106, 2012, in Chinese.
View at: Google Scholar
H. Cai, H. Liu, and M. Lu, “Fault-tolerant maximal local-connectivity on bubble-sort star graphs,” Discrete Applied Mathematics, vol. 181, pp. 33–40, 2015.
View at: Publisher Site | Google Scholar
W. Feng, Jirimutu, and S. Wang, “The nature diagnosability of wheel graph networks under the PMC model and model,” ARS Combinatoria, vol. 143, pp. 255–287, 2019.
View at: Google Scholar
Y. Wei and M. Xu, “The 1, 2-good-neighbor conditional diagnosabilities of regular graphs,” Applied Mathematics and Computation, vol. 334, pp. 295–310, 2018.
View at: Publisher Site | Google Scholar
J. Tu, Y. Zhou, and G. Su, “A kind of conditional connectivity of Cayley graphs generated by wheel graphs,” Applied Mathematics and Computation, vol. 301, pp. 177–186, 2017.
View at: Publisher Site | Google Scholar
M.-M. Gu and R.-X. Hao, “Reliability analysis of Cayley graphs generated by transpositions,” Discrete Applied Mathematics, vol. 244, pp. 94–102, 2018.
View at: Publisher Site | Google Scholar
X. Hu, Y. Tian, and J. Meng, “Super R^k-vertex-connectedness,” Applied Mathematics and Computation, vol. 339, pp. 812–819, 2018.
View at: Publisher Site | Google Scholar
X. Hu, W. Yang, Y. Tian, and J. Meng, “Equal relation between g-good-neighbor diagnosability under the PMC model and g-good-neighbor diagnosability under the model of a graph,” Discrete Applied Mathematics, vol. 262, pp. 96–103, 2019.
View at: Publisher Site | Google Scholar
Y. Wei and M. Xu, “Conditional diagnosability of Cayley graphs generated by wheel graphs under the PMC model,” Theoretical Computer Science, vol. 849, pp. 163–172, 2021.
View at: Publisher Site | Google Scholar
W. Feng and S. Wang, “The diagnosability of wheel networks with missing edges under the comparison model,” Mathematics, vol. 8, no. 10, p. 1818, 2020.
View at: Publisher Site | Google Scholar
W. Feng and S. Wang, “The 2-extra connectivity of wheel networks,” Mathematical Problems in Engineering, vol. 2020, Article ID 8910240, 5 pages, 2020.
View at: Publisher Site | Google Scholar
W. Feng and S. Wang, “Structure connectivity and substructure connectivity of wheel networks,” Theoretical Computer Science, vol. 850, pp. 20–29, 2021.
View at: Publisher Site | Google Scholar
F. Hou, Some New Results of the Wheel Networks and Bubble-Sort Star Networks, Northwest Normal University, Lanzhou, China, 2013, in Chinese.
W. Feng, J. Ren, C. Enhe, and S. Wang, “The 2-good-neighbor connectivity of wheel graph networks,” Utilitas Mathematica, vol. 116, pp. 139–167, 2020.
View at: Google Scholar
A. Ali, “A survey of antiregular graphs,” Contributions to Mathematics, vol. 1, pp. 67–79, 2020.
View at: Publisher Site | Google Scholar

Copyright

Copyright © 2021 Wei Feng and Shiying Wang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

PDF Download Citation

Download other formats

Order printed copies