Generating functions of tree graphs and Legendre transformation

Generating functions of tree graphs and Legendre transformation 

We associate a factor vm with each m-leg vertex and a factor — u with any internal or external interaction line (an external leg, for instance). The contribution of a graph is then defined as the product of all the factors associated with 

The generating function of the rootless graphs with m legs is given by 

In the same way, the generating function of the one-root graphs with (m — 1) legs is given by 

We shall now show that the Legendre transform B(y) of d(x) has a much simpler structure than /4(x). 

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