By V. K. Balakrishnan
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Additional info for Schaum's outline of theory and problems of graph theory
Corollary 22. Let S be a possible subgraph of Tn . With the notation above, the probability pS that S ⊂ Tn satisfies din S (i)! pS = i∈V −( S) i∈V +( S) 1 2i − 3 1+ t∈V / +( S) CS (t) 2t − 3 . 15 The case m = 1: plane-oriented recursive trees 31 Furthermore, 1 √ exp O 2 ij din S (i)! pS = i∈V ( S) ij∈E( S) i∈V −( S) CS (i)2 /i . 25) Proof. 22), and write pS = λn = λn /λ0 as the product of the factors appearing in these equations. The second statement follows by simple approximations: for all x ≥ 0 we have log(1 + x) = x + O(x2 ).
If c < cm then whp Gc has a component with Θ(n) vertices. If c > cm then whp Gc has no such component. Proof. We just give an outline, using the methods above. 19) is c rather than 0. Near the critical probability, the functions L and R will be small, and hence close to the solution of the linearized form of the equations. It is easy to solve these linearized equations; a non-zero solution exists if and only if c = cm , and one can deduce the result. 15 The case [0, 1]: plane-oriented recursive trees A simple special case of the BA model that has been considered in several papers is the m = 1 case, where each vertex sends a single edge to an earlier vertex, giving rise to a tree.
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