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

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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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