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| Main Authors: | , , , |
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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2404.17150 |
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| _version_ | 1866914119142080512 |
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| author | Sun, Yali Zhang, Mingzu Feng, Xing Yang, Xing |
| author_facet | Sun, Yali Zhang, Mingzu Feng, Xing Yang, Xing |
| contents | Reliability assessment of interconnection networks is critical to the design and maintenance of multiprocessor systems. The $(n, k)$-enhanced hypercube $Q_{n,k}$, as a variation of the hypercube $Q_{n}$, was proposed by Tzeng and Wei in 1991. As an extension of traditional edge-connectivity, $h$-extra edge-connectivity of a connected graph $G,$ $λ_h(G),$ is an essential parameter for evaluating the reliability of interconnection networks. This article intends to study the $h$-extra edge-connectivity of the $(n,2)$-enhanced hypercube $Q_{n,2}$. Suppose that the link malfunction of an interconnection network $Q_{n,2}$ does not isolate any subnetwork with no more than $h-1$ processors, the minimum number of these possible faulty links concentrates on a constant $2^{n-1}$ for each integer $\lceil\frac{11\times2^{n-1}}{48}\rceil \leq h \leq 2^{n-1}$ and $n\geq 9$. That is, for about $77.083\%$ of values where $h\leq2^{n-1},$ the corresponding $h$-extra edge-connectivity of $Q_{n,2}$, $λ_h(Q_{n,2})$, presents a concentration phenomenon. Moreover, the lower and upper bounds of $h$ mentioned above are both tight. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_17150 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A concentration phenomenon for $h$-extra edge-connectivity reliability analysis of enhanced hypercubes $Q_{n,2}$ with exponentially many faulty links Sun, Yali Zhang, Mingzu Feng, Xing Yang, Xing Combinatorics Discrete Mathematics Reliability assessment of interconnection networks is critical to the design and maintenance of multiprocessor systems. The $(n, k)$-enhanced hypercube $Q_{n,k}$, as a variation of the hypercube $Q_{n}$, was proposed by Tzeng and Wei in 1991. As an extension of traditional edge-connectivity, $h$-extra edge-connectivity of a connected graph $G,$ $λ_h(G),$ is an essential parameter for evaluating the reliability of interconnection networks. This article intends to study the $h$-extra edge-connectivity of the $(n,2)$-enhanced hypercube $Q_{n,2}$. Suppose that the link malfunction of an interconnection network $Q_{n,2}$ does not isolate any subnetwork with no more than $h-1$ processors, the minimum number of these possible faulty links concentrates on a constant $2^{n-1}$ for each integer $\lceil\frac{11\times2^{n-1}}{48}\rceil \leq h \leq 2^{n-1}$ and $n\geq 9$. That is, for about $77.083\%$ of values where $h\leq2^{n-1},$ the corresponding $h$-extra edge-connectivity of $Q_{n,2}$, $λ_h(Q_{n,2})$, presents a concentration phenomenon. Moreover, the lower and upper bounds of $h$ mentioned above are both tight. |
| title | A concentration phenomenon for $h$-extra edge-connectivity reliability analysis of enhanced hypercubes $Q_{n,2}$ with exponentially many faulty links |
| topic | Combinatorics Discrete Mathematics |
| url | https://arxiv.org/abs/2404.17150 |