The r-component connectivity κ r (G) of the non-complete graph G is the minimum number of vertices whose deletion results in a graph with at least r components. So, κ2 is the usual connectivity. In this paper, we determine the r-component connectivity of the hypercube Q n for r=2, 3, …, n+1, and we classify all the corresponding optimal solutions.

https://ir.nctu.edu.tw:443/bitstream/11536/15012/1/000298350400002.pdf

Component connectivity of the hypercubes

{2,3}-Extraconnectivities of hypercube-like networks

In this article, we study the periodicity, the boundedness and the global stab ility of the positive solutions of the following nonlinear difference equation

/pdf/dynamics-of-the-rational-difference-equation-4glq05tfe3.pdf

Dynamics of the Rational Difference Equation

Reliability evaluation of interconnection network is important to the design and maintenance of multiprocessor systems. The extra connectivity and the extra edge-connectivity are two important parameters for the reliability evaluation of interconnection networks. The ${\mbi {n}}$ -dimensional bijective connection network (in brief, BC network) includes several well known network models, such as, hypercubes, Mobius cubes, crossed cubes, and twisted cubes. In this paper, we explore the extra connectivity and the extra edge-connectivity of BC networks, and discuss the structure of BC networks with many faults. We obtain a sharp lower bound of ${{g}}$ -extra edge-connectivity of an ${\mbi {n}}$ -dimensional BC network for ${{n}} \geq 4$ and $1 \leq { {g}} \leq {2^{[{{{n}} \over 2}]}}$ . We also obtain a sharp lower bound of ${ {g}}$ -extra connectivity of an ${{n}}$ -dimensional BC network for ${{n}} \geq 4$ and $1 \leq { {g}} \leq 2{ {n}}$ which improves the result in [“Reliability evaluation of BC networks,” IEEE Trans. Computers, DOI: 10.1109/tc.2012.106.] for $1 \leq { {g}} \leq { {n}} - 3$ . Furthermore, we give a remark about exploring the ${ {g}}$ -extra edge-connectivity of BC networks for the more general ${\mbi {g}}$ , and we also characterize the structure of BC networks with many faulty nodes or links. As an application, we obtain several results on the ${\mbi {g}}$ -extra (edge-) connectivity and the structure of faulty networks on hypercubes, Mobius cubes, crossed cubes, and twisted cubes.

Reliability Evaluation of BC Networks in Terms of the Extra Vertex- and Edge-Connectivity

Component connectivity of hypercubes

Hypercube is one of the most popular topologies for connecting processors in multicomputer systems. In this paper we address the maximum order of a connected component in a faulty cube. The results established include several known conclusions as special cases. We conclude that the hypercube structure is resilient as it includes a large connected component in the presence of large number of faulty vertices.

Bill Chen

Papers

On the maximal connected component of hypercube with faulty vertices

On the recursive sequence xn=axn-1+bxn-2c+dxn-1xn-2

On the recursive sequence *1

Global attractivity in a recursive sequence

On the recursive sequence x n = (ax n-1 +bx n-2 ) / (c+dx n-1 x n-2 ).