  Dodgson polynomials appear in Schwinger parametric Feynman integrals and are
closely related to the well known Kirchhoff (or first Symanzik) polynomial. In
this article a new combinatorial interpretation and a generalisation of Dodgson
polynomials are provided. This leads to two new identities that relate large
sums of products of Dodgson polynomials to a much simpler expression involving
powers of the Kirchhoff polynomial. These identities can be applied to the
parametric integrand for quantum electrodynamics, simplifying it significantly.
This is worked out here in detail on the example of superficially renormalised
photon propagator Feynman graphs, but works much more generally.
