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Many of the network in the real world , such as computer networks and communication networks , etc., will be due to the communication line of the network or site failure , which led to the function of the network is reduced, or even the entire network failure. Therefore, analysis of the reliability of the existing network and the design of reliable the network has important theoretical significance and application value the abstract structure of the network can be described with a picture , in the study of network reliability , network is generally defined by a graph G = ( V, E) and vertex set V and edge set E to the interval [ 0,1 ] of two functions Φ : V → [ 0,1 ] and Ψ: E → [0,1] , and the values ??of these two functions are the vertices and edges out the probability of failure . In case of network reliability R (G, Φ, Ψ) is defined as each of the vertices and edges of the graph G Φ , Ψ is the failure probability function , Figure G to maintain the probability of communicating . Let G be the e edges of the n points of the network , it is assumed that the point does not fail while the probability of failure of each edge of q and independent of each other , then the edges of G VALID reliability : R e (G, q) = (?), wherein N i (G) said graph G contains the number of connectivity i edges generated subgraph , apparently N n -1 ( G) G n-1 edge connected spanning subgraph number , both the number of spanning tree , and therefore demand network optimization problem is transformed to calculate the number of Spanning Trees problem , in addition to looking τ- excellent Figure number of requirements given graph spanning tree is also very important . in this thesis, a synthetic graph spanning tree count and network reliability design . The first chapter introduces the main results to get involved in some of the basic concepts and terminology , current research and the paper . The second chapter study synthetic graph spanning tree counting problems , some new results . Chapter utilization side split set number two network reliability . Chapter IV of this paper , we discuss some further research .
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