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 n L with satisfying the property that each minimum P-conditional edgecut separates the K n L just into two components, such as L t -extra edge-connectivity, t-embedded edge-connectivity, cyclic edgeconnectivity, (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 fromBesides, we also obtain the exact values of h-extra edgeconnectivity and h-th isoperimetric edge-connectivity of hamming graph K n L for each h ≤ L ⌊ n 2 ⌋ . For the case L = 2, K n 2 = Qn 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,