Concentration behavior: 50 percent of h-extra edge connectivity of pentanary n-cube with exponential faulty edges

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.