Link fault tolerance of BC networks and folded hypercubes on h-extra r-component edge-connectivity

Parallel and distributed systems play a significant role in high-performance computing, prompting us to investigate qualitative and quantitative metrics to indicate the fault tolerance and vulnerability of systems. Consider the setup where there are large-scale link malfunctions that disconnect the network and result in various components, each with multiple processors. In this paper, we propose and study the h-extra r-component edge-connectivity of a connected graph , which is denoted by c lambda(h)(r)(G) and has not been addressed before. Let and be two positive integers with r >= 2. An edge subset F subset of E(G) is said to be an h-extra r-component edge-cut of , if any, G - F has at least components and every component of G - F has at least vertices. The cardinality of the minimum h-extra r-component edge-cut of is the -extra -component edge-connectivity of . Let be a positive integer with the decomposition h =Sigma(t)(t=0)2(ki), where k(t) > k(i+1), 0 <= i <= - t - 1. In this paper, we derive a lower bound for the exact value of h-extra 3-component edge-connectivity of BC networks B-n and demonstrate that it is tight for one member of B-n, hypercube Q(n), in the interval 1 <= h <=(2left perpendicularn/2right perpendicular-1) -1, n >= 4. This lower bound is also tight for the exact value of 2(c)-extra 3-component edge-connectivity of B-n with 0 <= c <= n - 2,n >= 4. Specifically, c lambda(h)(3)(Q(n)) = 2nh - Sigma(t)(t=0) (kt2ki+1) - Sigma(t)(t=0) (i2ki+2) - h for 1 <= h <= 2(left perpendicularn/2right perpendicular-1)-1, n >= 4 and c lambda(2l)(3)(B-n) = (2n-2c-1)(2c) for 0 <= c <= n - 2, n >= 4. Exact value of h-extra 3-component edge-connectivity of n-dimensional folded hypercube FQ(n), c lambda(h)(3)(FQ(n)) is 2(n + 1)h - Sigma(t)(t=0) (k)i(t2ki+1) - Sigma(t=0)i (i2kt+2) - h for 1 <= h <= 2(inverted right perpendicularn/2inverted left perpendicular-1) - 1, n >= 4, and that of 2(c)-extra 3-component edge-connectivity of FQ(n), c lambda(2c)(3)(FQ(n)), is (2n -2c + 1)2(c) for 0 <= c <= n -2, n >= 4.