Reliability measure of the n-th cartesian product of complete graph K4 on h-extra edge-connectivity

Abstract

As a measurement parameter of the reliability about interconnection networks of parallel and distributed systems, the h-extra edge-connectivity λh(G) is a better alternative compared with the classical Menger's theorem of the edge-connectivity. Recently, Li and Yang (2013) [7] determined the values of the h-extra edge-connectivity of hypercube Qn for each h≤2⌊n2⌋. Because of easy scalability, the interconnection networks based on cartesian product operation are extensively investigated. This paper focuses on the h-extra edge-connectivity of the n-th cartesian product of complete graph K4 with exponentially many faulty links. For a sufficiently large positive integer n, about 60 percent of positive integers h in the interval 1≤h≤2⋅4n−1 corresponding h-extra edge-connectivity of K4n, λh(K4n), presents a concentration phenomenon, that is, these exact values of λh(K4n) concentrate on 3⋅4n−1 and 4n for each ⌈3⋅4n−1/5⌉≤h≤4n−1 and ⌈6⋅4n−1/5⌉≤h≤2⋅4n−1, respectively. And the lower and upper bounds of h are sharp. Furthermore, the values of λh(K2n) also have this phenomenon. We obtain λh(K4n)=32λh(K22n)=3⋅4n−1 or λh(K4n)=2λh(K22n)=4n in the subintervals where the concentration phenomenon occurs simultaneously.