{"title":"Concentration behavior: 50 percent of h-extra edge connectivity of pentanary n-cube with exponential faulty edges","authors":"Tengteng Liang, Mingzu Zhang, Sufang Liu","doi":"10.1007/s10878-023-01098-3","DOIUrl":null,"url":null,"abstract":"<p>Edge disjoint paths have a closed relationship with edge connectivity and are anticipated to garner increased attention in the study of the reliability and edge fault tolerance of a readily scalable interconnection network. Note that this interconnection network is always modeled as a connected graph <i>G</i>. The minimum of some of modified edge-cuts of a connected graph <i>G</i>, also known as the <i>h</i>-extra edge-connectivity of a graph <i>G</i> (<span>\\(\\lambda _{h}(G)\\)</span>), is defined as the maximum number of the edge disjoint paths connecting any two disjoint connected subgraphs with <i>h</i> vertices in the graph <i>G</i>. From the perspective of edge-cut, the smallest cardinality of a collection of edges, whose removal divides the whole network into several connected subnetworks having at least <i>h</i> vertices, is the <i>h</i>-extra edge-connectivity of the underlying topological architecture of an interconnection network <i>G</i>. This paper demonstrates that the <i>h</i>-extra edge-connectivity of the pentanary <i>n</i>-cube (<span>\\(\\lambda _{h}(K_{5}^{n})\\)</span>) appears a concentration behavior for around 50 percent of <span>\\(h\\le \\lfloor 5^{n}/2\\rfloor \\)</span> as <i>n</i> approaches infinity. Let <span>\\(e=1\\)</span> for <i>n</i> is even and <span>\\(e=0\\)</span> for <i>n</i> is odd. It mainly concentrates on the value <span>\\([4g(\\lceil \\frac{n}{2}\\rceil -r)-g(g-1)]5^{\\lfloor \\frac{n}{2}\\rfloor +r}\\)</span> for <span>\\(g5^{\\lfloor \\frac{n}{2}\\rfloor +r}-\\lfloor \\frac{[(g-1)^{2}+1]5^{2r+e}}{3}\\rfloor \\le h\\le g5^{\\lfloor \\frac{n}{2}\\rfloor +r}\\)</span>, where <span>\\(r=1, 2,\\cdots , \\lceil \\frac{n}{2}\\rceil -2\\)</span>, <span>\\(g\\in \\{1, 2,3,4\\}\\)</span>; <span>\\(r=\\lceil \\frac{n}{2}\\rceil -1\\)</span>, <span>\\(g\\in \\{1,2\\}\\)</span>. Furthermore, it is shown that the above upper bound and lower bound of <i>h</i> are sharp.</p>","PeriodicalId":50231,"journal":{"name":"Journal of Combinatorial Optimization","volume":null,"pages":null},"PeriodicalIF":0.9000,"publicationDate":"2024-01-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Combinatorial Optimization","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s10878-023-01098-3","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS","Score":null,"Total":0}
引用次数: 0
Abstract
Edge disjoint paths have a closed relationship with edge connectivity and are anticipated to garner increased attention in the study of the reliability and edge fault tolerance of a readily scalable interconnection network. Note that this interconnection network is always modeled as a connected graph G. The minimum of some of modified edge-cuts of a connected graph G, also known as the h-extra edge-connectivity of a graph G (\(\lambda _{h}(G)\)), is defined as the maximum number of the edge disjoint paths connecting any two disjoint connected subgraphs with h vertices in the graph G. From the perspective of edge-cut, the smallest cardinality of a collection of edges, whose removal divides the whole network into several connected subnetworks having at least h vertices, is the h-extra edge-connectivity of the underlying topological architecture of an interconnection network G. This paper demonstrates that the h-extra edge-connectivity of the pentanary n-cube (\(\lambda _{h}(K_{5}^{n})\)) appears a concentration behavior for around 50 percent of \(h\le \lfloor 5^{n}/2\rfloor \) as n approaches infinity. Let \(e=1\) for n is even and \(e=0\) for n is odd. It mainly concentrates on the value \([4g(\lceil \frac{n}{2}\rceil -r)-g(g-1)]5^{\lfloor \frac{n}{2}\rfloor +r}\) for \(g5^{\lfloor \frac{n}{2}\rfloor +r}-\lfloor \frac{[(g-1)^{2}+1]5^{2r+e}}{3}\rfloor \le h\le g5^{\lfloor \frac{n}{2}\rfloor +r}\), where \(r=1, 2,\cdots , \lceil \frac{n}{2}\rceil -2\), \(g\in \{1, 2,3,4\}\); \(r=\lceil \frac{n}{2}\rceil -1\), \(g\in \{1,2\}\). Furthermore, it is shown that the above upper bound and lower bound of h are sharp.
边缘不相交路径与边缘连通性之间存在闭合关系,预计在研究可随时扩展的互连网络的可靠性和边缘容错性时会受到越来越多的关注。需要注意的是,这种互连网络总是被建模为一个连通图 G。连通图 G 的一些修改过的边切的最小值,也称为图 G 的 h 外边连通性(\(\lambda _{h}(G)\) ),定义为连接图 G 中具有 h 个顶点的任意两个互不相交的连通子图的边脱节路径的最大数目。从边缘切割的角度来看,一个边缘集合的最小心数,即互联网络 G 的底层拓扑结构的 h 外边缘连通性,去除该边缘集合可将整个网络划分为至少有 h 个顶点的多个连通子网络。本文证明,当 n 接近无穷大时,五元 n 立方体(\(\lambda _{h}(K_{5}^{n})\)的 h-extra edge-connectivity 出现了集中行为,约为\(h\le \lfloor 5^{n}/2\rfloor \)的 50%。当 n 为偶数时,让 \(e=1\);当 n 为奇数时,让 \(e=0\)。它主要集中在 \([4g(\lceil \frac{n}{2}\rceil -r)-g(g-1)]5^{\lfloor \frac{n}{2}\rfloor +r}\) 的值上。\frac{[(g-1)^{2}+1]5^{2r+e}}{3}\rfloor \le h\le g5^{lfloor \frac{n}{2}\rfloor +r}/)、其中(r=1,2,cdots, \lceil \frac{n}{2}\rceil -2),(gin \{1,2,3,4});\r=\lceil\frac{n}{2}\rceil -1\),\(g\in\{1,2\}).此外,还证明了上述 h 的上界和下界都很尖锐。
期刊介绍:
The objective of Journal of Combinatorial Optimization is to advance and promote the theory and applications of combinatorial optimization, which is an area of research at the intersection of applied mathematics, computer science, and operations research and which overlaps with many other areas such as computation complexity, computational biology, VLSI design, communication networks, and management science. It includes complexity analysis and algorithm design for combinatorial optimization problems, numerical experiments and problem discovery with applications in science and engineering.
The Journal of Combinatorial Optimization publishes refereed papers dealing with all theoretical, computational and applied aspects of combinatorial optimization. It also publishes reviews of appropriate books and special issues of journals.