  We investigate the relationship between the universal topological polynomials
for graphs in mathematics and the parametric representation of Feynman
amplitudes in quantum field theory. In this first paper we consider translation
invariant theories with the usual heat-kernel-based propagator. We show how the
Symanzik polynomials of quantum field theory are particular multivariate
versions of the Tutte polynomial, and how the new polynomials of noncommutative
quantum field theory are particular versions of the Bollob\'as-Riordan
polynomials.
