  The Feynman rules assign to every graph an integral which can be written as a
function of a scaling parameter L. Assuming L for the process under
consideration is very small, so that contributions to the renormalizaton group
are small, we can expand the integral and only consider the lowest orders in
the scaling. The aim of this article is to determine specific combinations of
graphs in a scalar quantum field theory that lead to a remarkable
simplification of the first non-trivial term in the perturbation series. It
will be seen that the result is independent of the renormalization scheme and
the scattering angles. To achieve that goal we will utilize the parametric
representation of scalar Feynman integrals as well as the Hopf algebraic
structure of the Feynman graphs under consideration. Moreover, we will present
a formula which reduces the effort of determining the first-order term in the
perturbation series for the specific combination of graphs to a minimum.
