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

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.