  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 $n$-dim-ensional bijective connection networks, which
contains hypercubes, twisted cubes, crossed cubes, M\"obius cubes, locally
twisted cubes and so on. Our results improve several previous results on this
topic.
