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

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 ( $$\lambda _{h}(G)$$ λ 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 ( $$\lambda _{h}(K_{5}^{n})$$ λ h ( K 5 n ) ) appears a concentration behavior for around 50 percent of $$h\le \lfloor 5^{n}/2\rfloor $$ h ≤ ⌊ 5 n / 2 ⌋ as n approaches infinity. Let $$e=1$$ e = 1 for n is even and $$e=0$$ e = 0 for n is odd. It mainly concentrates on the value $$[4g(\lceil \frac{n}{2}\rceil -r)-g(g-1)]5^{\lfloor \frac{n}{2}\rfloor +r}$$ [ 4 g ( ⌈ n 2 ⌉ - r ) - g ( g - 1 ) ] 5 ⌊ n 2 ⌋ + r for $$g5^{\lfloor \frac{n}{2}\rfloor +r}-\lfloor \frac{[(g-1)^{2}+1]5^{2r+e}}{3}\rfloor \le h\le g5^{\lfloor \frac{n}{2}\rfloor +r}$$ g 5 ⌊ n 2 ⌋ + r - ⌊ [ ( g - 1 ) 2 + 1 ] 5 2 r + e 3 ⌋ ≤ h ≤ g 5 ⌊ n 2 ⌋ + r , where $$r=1, 2,\cdots , \lceil \frac{n}{2}\rceil -2$$ r = 1 , 2 , ⋯ , ⌈ n 2 ⌉ - 2 , $$g\in \{1, 2,3,4\}$$ g ∈ { 1 , 2 , 3 , 4 } ; $$r=\lceil \frac{n}{2}\rceil -1$$ r = ⌈ n 2 ⌉ - 1 , $$g\in \{1,2\}$$ g ∈ { 1 , 2 } . Furthermore, it is shown that the above upper bound and lower bound of h are sharp.