Research On Reliability Of The Hypercube-Like Networks

Reliability is the prerequisite to ensure the normal operation of the networks,so the research on reliability is very important to measure the performance of the networks.The edge connectivity is a crucial parameter when evaluating the reliability of a network.As the improvement and generalization of the edge connectivity,the component edge connectivity and the conditional edge-fault-tolerance about the SM-λ property are powerful parameters for evaluating the reliability of the network.Let G be non-complete graph and S?(E（G）,the m-component edge cut is edge set S such that G-S has at least m components.The m-component edge connectivity cλ_m（G）is the minimum size of the m-component edge cut of G.For any pair of vertices u and v of the connected graph G,if they are connected by min{degG（u）,degG（v）} edgedisjoint paths,then G is strong Menger edge connected（SM-λ for short）.Let S ? E（G）and satisfy the following condition:|S|≤m and δ（G-S）≥r.The conditional edgefault-tolerance about the SM-λ property of G,written sm_λ^r（G）,is the maximum value of m such that G-S is still SM-λ.Hypercube-like networks are obtained by manipulating some pairs of edges in hypercubes,which contain several famous interconnection networks such as twisted cubes,M?bius cubes,crossed cubes etc.The section 2 determine the（m+1）-component edge connectivity of n-dimensional hypercube-like networks cλ_m+1（G_n）=nm-em for m≤2[n/2],n≥8.Previously,the exact value of sm_λ^r（G）on some well-known networks was given only when r≤2,and for r≥3,only the lower bounds of sm_λ^r（G）was determined.The section 3 firstly determine the exact value of sm_λ^r（G_n）on n-dimensional hyp ercube-like networks for a general r,that is,sm_λ^r（G_n）=2^r（n-r）-n,where n≥3 and 1≤r≤n-2.