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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 <tex>$\mathcal{P}$</tex> -conditional edge-connectivity. With the help of edge isoperimetric problem&#x0027;s method in combinatorics, this paper mainly offers a novel and unified view to investigate the <tex>$\mathcal{P}$</tex> -conditional edge-connectivities of hamming graph <tex>$K_{L}^{n}$</tex> with satisfying the property that each minimum <tex>$\mathcal{P}$</tex> -conditional edge-cut separates the <tex>$K_{L}^{n}$</tex> just into two components, such as <tex>$L^{t}$</tex> -extra edge-connectivity, t-embedded edge-connectivity, cyclic edge-connectivity, <tex>$(L-1)t$</tex> -super edge-connectivity, <tex>$(L-1)t$</tex> -average edge-connectivity and <tex>$L^{t}$</tex> -th isoperimetric edge-connectivity. They share the same values in form of <tex>$(L-1)(n-t)L^{t}$</tex> (except for cyclic edge-connectivity), which equals to the minimum number of links-faulty resulting in an L-ary-n-dimensional sub-layer from <tex>$K_{L}^{n}$</tex>. Besides, we also obtain the exact values of h-extra edge-connectivity and h-th isoperimetric edge-connectivity of hamming graph <tex>$K_{L}^{n}$</tex> for each <tex>$h\leq L^{\lfloor\frac{n}{2}\rfloor}$</tex>. For the case <tex>$L=2, K_{2}^{n}=Q_{n}$</tex> 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, M&#x00F6;bius cubes, locally twisted cubes and so on. Our results improve several previous results on this topic.

A Novel View: Edge is Operimetric Methods and Reliability Evaluation of Several kinds of Conditional Edge-connectivity of Interconnection Networks