A novel view: edge isoperimetric methods and reliability evaluation of several kinds of conditional edge-connectivity of interconnection networks
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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 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 𝒫-conditional edge-connectivities of hamming graph K_L^n with satisfying the property that each minimum 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≤ L^⌊n/2⌋. 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-dim-ensional 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.
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