A novel view: edge isoperimetric methods and reliability evaluation of several kinds of conditional edge-connectivity of interconnection networks

TL;DR

This paper introduces a unified combinatorial approach using edge isoperimetric methods to evaluate various conditional edge-connectivities in interconnection networks, providing exact values and improving previous results.

Contribution

It offers a novel, unified framework for analyzing multiple types of conditional edge-connectivity in interconnection networks, including hypercubes and related structures.

Findings

Exact values of multiple conditional edge-connectivities for hamming graphs.

Unified formula for different edge-connectivities, except cyclic edge-connectivity.

Results applicable to a broad class of networks including hypercubes and twisted cubes.

Abstract

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.