  In order to use the Gaussian representation for propagators in Feynman
amplitudes, a representation which is useful to relate string theory and field
theory, one has to prove first that each $\alpha$- parameter (where $\alpha$ is
the parameter associated to each propagator in the $\alpha$-representation of
the Feynman amplitudes) can be replaced by a constant instead of being
integrated over and second, prove that this constant can be taken equal for all
propagators of a given graph. The first proposition has been proven in one
recent letter when the number of propagators is infinite. Here we prove the
second one. In order to achieve this, we demonstrate that the sum over the
weighted spanning trees of a Feynman graph $G$ can be factorized for disjoint
parts of $G$. The same can also be done for cuts on $G$, resulting in a
rigorous derivation of the Gaussian representation for super-renormalizable
scalar field theories. As a by-product spanning trees on Feynman graphs can be
used to define a discretized functional space.
