On the combinatorics of Feynman graphs. I

As a preliminary step towards a rigorous proof ofDyson's conjectures on the combinatorics of Feynman graphs, the following lemma is proved rigorously in an explicit and elementary way: The generating functional of the time-orderedGreen's functions in perturbation theory is an exponential, the exponent of which is a sum of contributions from connected graphs only, with no contributions from vacuum graphs. The model used to demonstrate this is a scalar hermitean fieldA(x) with anA ⁴ selfcoupling.