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 article 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 special versions of the Bollobás–Riordan polynomials.
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Thomas Krajewski, Vincent Rivasseau, Adrian Tanasă, Zhituo Wang, Topological graph polynomials and quantum field theory<br> Part I: heat kernel theories. J. Noncommut. Geom. 4 (2010), no. 1, pp. 29–82DOI 10.4171/JNCG/49