Topological graph polynomials and quantum field theory

We define a new topological polynomial extending the Bollobás-Riordan one, which obeys a four-term reduction relation of the deletion/contraction type and has a natural behavior under partial duality. This allows to write down a completely explicit combinatorial evaluation of the polynomials, occurring in the parametric representation of the non-commutative Grosse-Wulkenhaar quantum field theory. An explicit solution of the parametric representation for commutative field theories based on the Mehler kernel is also provided. 1 ha l-0 04 44 33 0, v er si on 1 6 Ja n 20 10