Reliability Analysis of Crossed Cube Networks on Degree

Crossed cubes network is a kind of interconnection structure as a basis for distributed memory parallel computer architecture. Reliability takes an important role in fault tolerant computing on multiprocessor systems. Connectivity is a vital metric to explore fault tolerance and reliability of network structure based on a graph model. Let ( , ) G V E = be a connected graph. The k-conditional edge connectivity ( ) k G λ is the cardinality of the minimum edge cuts F , if any, whose deletion disconnects G and each component of G F − has property of minimum degree k δ ≥ . The k-conditional connectivity ( ) k G κ can be defined similarly. In this paper, we determine the kconditional (edge) connectivity of crossed cubes n CQ for small k. And we also prove other properties of n CQ .


Introduction
With the development of VLSI technology and software technology, multiprocessor systems with hundreds of thousands of processors have become available.
With the continuous increase in the size of multiprocessor systems, the complexity of a system can adversely affect its fault tolerance and reliability.To the design and maintenance purpose of multiprocessor systems, appropriate measures of reliability should be found.Let G be a connected graph and P be graph-theoretic property.δ ≥ .The k-conditional connectivity ( ) k G κ can be obtained similarly.In recent years, numerous results about many kind of connectivities on networks have been reported [4]- [20].

Let
( , ) X Y the set of edges of G with one end in X and the other in Y .For graph-theoretical terminology and notation not defined here we follow [21].All graphs considered in this paper are simple, finite and undirected.
Two binary strings K , the complete graph with labels 0 and 1.For 2) for From the definition, we can see that each vertex of n CQ with a leading 0 bit has exactly one neighbor with a leading 1 and vice versa.It is an n-regular graph.
In fact, some pairs of parallel edges are changed to some pairs of cross edges.
Furthermore, n CQ can be obtained by adding a perfect matching M between  , ) The crossed cube is an attractive alternative to hypercubes n Q .The diameter of n CQ is approximately half that of n Q .For more references, we can see [23]- [29] (Figure 1).
In this paper, we obtain that: 2 ( ) 4 8( 4) ≥ , and we also prove other properties of n CQ .

Conditional Connectivity of Crossed Cubes
The crossed cube n CQ has an important property as follows.
Lemma By symmetry, we assume that  L. T.
, then the one common neighbor is in 0 1 n CQ − , and the other one is in V CQ − , then the two common neighbors are in Theorem 2.6.
is connected from the inductive hypothesis.We will show that every vertex of Let H be an any component of

Conclusion
The conditional connectivity is a generalization of classical connectivity of graphs.We determined the r-conditional degree connectivity of n CQ for the small r.In the future, we will study other properties of crossed cubes.

A
network is often modeled by a graph ( , ) G V E = with the vertices representing nodes such as processors or stations, and the edges representing links between the nodes.One fundamental consideration in the design of net-L.T. Guo DOI: 10.4236/jcc.2018.61014130 Journal of Computer and Communications works is reliability [1] [2].An edge cut of a connected graph G is a set of edges whose removal disconnects G .The edge connectivity ( ) G λ or connectivity ( ) G κ of G is the minimum cardinality of an edge cut or vertex cut S of G .The edge connectivity ( ) G λ or connectivity ( ) G κ is an important feature determining reliability and fault-tolerance of the network.In the definitions of ( ) G λ or ( ) G κ , no restrictions are imposed on the components of G S − .To compensate for this short coming, it would seem natural to generalize the notion of the classical connectivity by imposing some conditions or restrictions on the components of G S − .Following this idea, conditional connectivity were proposed in [3] by Harary.
neighbors must be obtained by adding the perfect matching M .Note that every vertex of 0 1 n CQ − has only one neighbor in 1 1 n CQ − and vice versa.Then we ob- tain the result.Corollary 2.2.For any two vertices , ( , ) 2 d x y = , then they have at most two common neighbors;2) if ( , ) 2 d x y ≠ , then they do not have common neighbors.
x be a some vertex of C .Because of | | 4 9 F n ≤ − , at least one vertex of CQ , and similar to Lemma 2.1, we have Lemma 2.4.Let x and y be any two vertices of ( Guo DOI: 10.4236/jcc.2018.61014132 Journal of Computer and Communications n