## Graph theoretic reliability analysis for the Boolean $$n$$ cube networks.(English)Zbl 0656.94027

It is well known that there are several papers which deal with the Boolean n-cube network and applications. However, no theoretical work has been done in calculating the network reliability of the Boolean $$n$$-cube network. In this paper, two graph theoretic results concerning about the problem of the Boolean $$n$$-cube network reliability are presented. First, a simple formula for the number of spanning trees of the Boolean $$n$$-cube network is derived. As a result, the reliability function for large failure rate can be readily computed. Second, the Boolean $$n$$-cube network is proved to have the super line-connectivity (see Definition 6) property. Thus the number of line disconnecting sets (a set of lines whose removal results in a disconnected or trivial graph) of order $$\lambda$$ (see Definition 3) for the Boolean $$n$$-cube network is equal to $$2^ n$$.

### MSC:

 94C15 Applications of graph theory to circuits and networks
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