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Reliability analysis of the pentanary n-cube based on h-extra edge-connectivity with a concentration behavior

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Abstract

The h-extra edge-connectivity is an important parameter for the reliability evaluation and fault tolerance analysis of the easily scalable interconnection networks of parallel and distributed systems. The h-extra edge-connectivity of the topological structure of an interconnection network G, denoted by \(\lambda _{h}(G)\), is the minimum cardinality of a set of link malfunctions whose deletion disconnects G and each remaining component has at least h processors. In this paper, for the integer \(n\ge 3\), we find that the h-extra edge-connectivity of n-dimensional pentanary cube (obtained by the n-th Cartesian product of \(K_{5}\)), denoted by \(\lambda _{h}(K_{5}^{n})\), presents a concentration behavior on the value \(4\times 5^{n-1}\) (resp. \(6\times 5^{n-1}\)) for some exponentially large enough h: \(\lceil \frac{2\times 5^{n-1}}{3}\rceil \le h\le 5^{n-1}\) (resp. \(\lceil \frac{4\times 5^{n-1}}{3}\rceil \le h\le 2\times 5^{n-1}\)). That is, for about 40.00 percent of \(1\le h\le \lfloor 5^{n}/2\rfloor \), the exact values of the h-extra edge-connectivity of n-dimensional pentanary cube are either \(4\times 5^{n-1}\) or \(6\times 5^{n-1}\).

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Notes

  1. when \(s=0\), \(ex_{m}(K_{5}^{n})=4a_{0}b_{0}5^{b_{0}}+2I_{a_{0}}5^{b_{0}}\)

References

  1. Abd-El-Barr M, Gebali F (2014) Reliability analysis and fault tolerance for hypercube multi-computer networks. Inf Sci 276:295–318

    Article  MathSciNet  Google Scholar 

  2. Balbuena C, Marcote X (2013) The \(k\)-restricted edge-connectivity of a product of graphs. Discret Appl Math 161(1):52–59

    Article  MathSciNet  Google Scholar 

  3. Chang N-W, Tsai C-Y, Hsieh S-Y (2014) On 3-extra connectivity and 3-extra edge connectivity of folded hypercubes. IEEE Trans Comput 63(6):1594–1600

    Article  MathSciNet  Google Scholar 

  4. Chen X, Liu J, Meng JX (2009) The restricted arc connectivity of cartesian product digraphs. Inf Process Lett 109(21–22):1202–1205

    Article  MathSciNet  Google Scholar 

  5. F\(\grave{a}\)brega J, Fiol MA (1994) Extra connectivity of graphs with large girth. Discrete Math 127(1–3):163–170

  6. F\(\grave{a}\)brega J, Fiol MA (1996) On the extra connectivity of graphs. Discrete Math 155(1–3):49–57

  7. Ghozati SA, Wasserman HC (1999) The \(k\)-ary \(n\)-cube network: modeling, topological properties and routing strategies. Comput Electr Eng 25(3):155–168

    Article  Google Scholar 

  8. Guo LT, Zhang MZ, Zhai SH, Xu LQ (2021) Relation of extra edge connectivity and component edge connectivity for regular networks. Int J Found Comput Sci 32(2):137–149

    Article  MathSciNet  Google Scholar 

  9. Harary F (1983) Conditional connectivity. Networks 13(3):347–357

    Article  MathSciNet  Google Scholar 

  10. Jo S, Park J-H, Chwa K-Y (2013) Paired many-to-many disjoint path covers in faulty hypercubes. Theoret Comput Sci 513:1–24

    Article  MathSciNet  Google Scholar 

  11. Kle\(\check{s}\check{c}\) M (2001) The crossing numbers of cartesian products of paths with 5-vertex graphs. Discrete Math 233(1–3):353–359

  12. Kle\(\check{s}\check{c}\) M (2001) On the crossing numbers of products of stars and graphs of order five. Graphs Combinatorics 17:289–294

  13. Kle\(\check{s}\check{c}\) M (1999) The crossing numbers of \(K_{5}\times P_{n}\). Tatra Mountains Mathematical Publications 18:63–68

  14. Li H, Yang WH (2013) Bounding the size of the subgraph induced by \(m\) vertices and extra edge-connectivity of hypercubes. Discret Appl Math 161:2753–2757

    Article  MathSciNet  Google Scholar 

  15. Li X-Y, Lin W, Liu X, Lin C-K, Pai K-J, Chang J-M (2021) Completely independent spanning trees on BCCC data center networks with an application to fault-tolerant routing. IEEE Trans Parallel Distrib Syst 33(8):1939–1952

    Article  Google Scholar 

  16. Liu XM, Meng JX (2021) The \(k\)-restricted edge-connectivity of the data center network DCell. Appl Math Comput 396. https://doi.org/10.1016/j.amc.2020.125941

  17. Lü M, Chen G-L, Xu X-R (2009) On super edge-connectivity of product graphs. Appl Math Comput 207(2):300–306

    MathSciNet  MATH  Google Scholar 

  18. L\(\ddot{u}\) SX, Huang YQ (2008) On the crossing numbers of \(K_{5}\times S_{n}\). J Math Res Exposition 28(3):445–459

  19. Montejano LP, Sau I (2017) On the complexity of computing the \(k\)-restricted edge-connectivity of a graph. Theoret Comput Sci 662:31–39

    Article  MathSciNet  Google Scholar 

  20. Qin YY, Xiong ZP, Wang JY (2010) On super 3-restricted edge connectivity of regular strong product graphs with girth at least four. In: International Conference on Networking and Digital Society, vol 1, pp 542–544

  21. Saraf JB, Borse YM, Mundhe G (2020) On conditional connectivity of the cartesian product of cycles. Discuss Math Graph Theory. https://doi.org/10.7151/dmgt.2348

    Article  Google Scholar 

  22. Wei YL, Li R-H, Yang WH (2021) The \(g\)-extra edge-connectivity of balanced hypercubes. J Interconnection Netw 21(4). https://doi.org/10.1142/S0219265921420081

  23. Xu LQ, Zhou SM (2021) An \(O({\log }_2(N))\) algorithm for reliability assessment of augmented cubes based on \(h\)-extra edge-connectivity. J Supercomput,1–13. https://doi.org/10.1007/s11227-021-04129-0

  24. Xu JM, Zhu Q, Hou MX, Zhou T (2005) On restricted edge connectivity and extra edge connectivity of hypercubes and folded hypercubes. J Shanghai Jiaotong Univ (Chin Ed) 2:203–207

    MATH  Google Scholar 

  25. Yang WH, Li HQ (2014) Reliability evaluation of BC networks in terms of the extra vertex- and edge-connectivity. IEEE Trans Comput 63(10):2540–2548

    Article  MathSciNet  Google Scholar 

  26. Yang WH, Li H (2014) On reliability of the folded hypercubes in terms of the extra edge-connectivity. Inf Sci 272:238–243

    Article  MathSciNet  Google Scholar 

  27. Ye L-C, Liang J-R, Lin H-X (2016) A fast pessimistic diagnosis algorithm for hypercube-like networks under the comparison model. IEEE Trans Comput 65(9):2884–2888

    Article  MathSciNet  Google Scholar 

  28. Yu ZC, Xu LQ, Yin SS, Guo LT (2022) Super vertex (edge)-connectivity of varietal hypercube. Symmetry 14(2). https://doi.org/10.3390/sym14020304

  29. Zhang QF, Xu LQ, Zhou SM, Guo LT (2020) Reliability analysis of subsystem in balanced hypercubes. IEEE Access 8:26478–26486

    Article  Google Scholar 

  30. Zhang QF, Xu LQ, Yang WH (2021) Reliability analysis of the augmented cubes in terms of the extra edge-connectivity and the component edge-connectivity. J Parallel Distrib Comput 147:124–131

    Article  Google Scholar 

  31. Zhu Q, Xu JM (2006) On restricted edge connectivity and extra edge connectivity of hypercubes and folded hypercubes. J Univ Sci Technol China 36(3):249–253

    MATH  Google Scholar 

  32. Zhang MZ, Meng JX, Yang WH, Tian YZ (2014) Reliability analysis of bijective connection networks in terms of the extra edge-connectivity. Inf Sci 279:374–382

    Article  MathSciNet  Google Scholar 

  33. Zhang MZ, Zhang LZ, Feng X, Lai HJ (2018) An \(O({\log }_2(N))\) algorithm for reliability evaluation of \(h\)-extra edge-connectivity of folded hypercubes. IEEE Trans Reliab 67:297–307

    Article  Google Scholar 

  34. Zhang MZ (2018) Edge isopermetric problem on graphs and the related applications. University of Xiamen, Xiamen, pp 68–77

    Google Scholar 

  35. Zhang MZ, Zhang LZ, Feng X (2016) Reliability measures in relation to the \(h\)-extra edge-connectivity of folded hypercubes. Theoret Comput Sci 615:71–77

    Article  MathSciNet  Google Scholar 

  36. Zhao WS, Ou JP (2013) On restricted edge-connectivity of lexicographic product graphs. Int J Comput Math 91(8):1618–1626

    Article  MathSciNet  Google Scholar 

  37. Zheng WP, Lin XH, Yang YS, Cui C (2007) On the crossing number of \(K_{m}\times P_{n}\). Graphs Combinatorics 23:327–336

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. College of Mathematics and System Sciences, Xinjiang University, Urumqi, 830046, China

    Tengteng Liang & Mingzu Zhang

  2. Department of Electrical and Computer Engineering, The University of Mississippi, Oxford, MS, 38677, USA

    Xing Yang

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  1. Tengteng Liang
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  2. Mingzu Zhang
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Correspondence to Mingzu Zhang.

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This work was supported by Science and Technology Project of Xinjiang Uygur Autonomous Region (No. 2020D01C069), National Natural Science Foundation of China (No. 12101528), Doctoral Startup Foundation of Xinjiang University (No. 62031224736), Tianchi Ph.D Program (Grant No. tcbs201905) and Xinjiang Key Laboratory of Applied Mathematics, No. XJDX1401.

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Liang, T., Zhang, M. & Yang, X. Reliability analysis of the pentanary n-cube based on h-extra edge-connectivity with a concentration behavior. J Supercomput 78, 15504–15531 (2022). https://doi.org/10.1007/s11227-022-04489-1

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