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Consider a network represented by a graph G with a probability of vertex failurep and a probability of edge failure q. The reliability of G, denoted by R（G）, is theconnected probability of all vertices in G. The main aim of network reliability design is tofind networks with maximum network reliability with given number of vertices and edges.Results on network reliability design are very important and helpful for designing reliablenetwork.Network reliability can be roughly divided into following: （1） reliability of networkwith edge failure; （2） reliability of network with vertex failure; and （3） mixed reliabilityof network with both vertex and edge failure. lots of authors be study the network reli-ability from some parameters. In this paper, we mainly investigate mixing local networkreliability.In chapter 1, we introduce the subtrees and network reliability status of research anddevelopment, as well as some of the major results already.In chapter 2, we introduce some of the basic concepts and terminology of the thesisand in Section II of this chapter gives the main results of the use of part of the lemmaand formulas.In Chapter 3, we mainly investigate the relationship between the mixed reliability ofnetwork with both vertex and edge failure the number of subtrees, we get1) when p→0, q→1 and p + q < 1, p = o（1-q）ifτ（G） >τ（G′）, then,R（G; p, q） > R（G′; p, q）.2) when p→0, q→1 and p+q = 1if S（G） > S（G′）, then, R（G; p, q） > R（G′; p, q）.3) when p→0, q→1 and p + q > 1, 1-q = o（p）if S3（G） > S3（G′）, then,R（G; p, q） > R（G′; p, q）.In Chapter 4, we mainly intrestigate the relationship them many of the subtrees andthe structure of unicyclic graph. we give1) Among unicyclic graphs, the maximum substrees is C3 K1,n-3, and S(C3 K1,n-3) =6 2n-3 + n; 2) Among unicyclic graphs, the minimum substrees is C3 Pn-2, and3) when p→0, q→1 and p + q = 1, locally optimal of the unicyclic graph isWe conclude in Chapter 5 of the thesis by discussing some problems for furtherreseach.
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