A Novel View: Edge is Operimetric Methods and Reliability Evaluation of Several kinds of Conditional Edge-connectivity of Interconnection Networks
Mingzu Zhang, Zhaoxia Tian, Lianzhu Zhang · 2021 IEEE 21st International Conference on Software Quality, Reliability and Security Companion (QRS-C) · 2021
Reliability evaluation and fault tolerance of an interconnection network of some parallel and distributed systems are discussed separately under various link-faulty hypotheses in terms of different$\mathcal{P}$-conditional edge-connectivity. With the help of edge isoperimetric problem's method in combinatorics, this paper mainly offers a novel and unified view to investigate the$\mathcal{P}$-conditional edge-connectivities of hamming graph$K_{L}^{n}$with satisfying the property that each minimum$\mathcal{P}$-conditional edge-cut separates the$K_{L}^{n}$just into two components, such as$L^{t}$-extra edge-connectivity, t-embedded edge-connectivity, cyclic edge-connectivity,$(L-1)t$-super edge-connectivity,$(L-1)t$-average edge-connectivity and$L^{t}$-th isoperimetric edge-connectivity. They share the same values in form of$(L-1)(n-t)L^{t}$(except for cyclic edge-connectivity), which equals to the minimum number of links-faulty resulting in an L-ary-n-dimensional sub-layer from$K_{L}^{n}$. Besides, we also obtain the exact values of h-extra edge-connectivity and h-th isoperimetric edge-connectivity of hamming graph$K_{L}^{n}$for each$h\leq L^{\lfloor\frac{n}{2}\rfloor}$. For the case$L=2, K_{2}^{n}=Q_{n}$is n-dimensional hypercube. Our results can be applied to more generalized class of networks, called n-dimensional bijective connection networks, which contains hypercubes, twisted cubes, crossed cubes, Möbius cubes, locally twisted cubes and so on. Our results improve several previous results on this topic.