Ribbon graphs embedded on a Riemann surface provide a useful way to describe
the double line Feynman diagrams of large N computations and a variety of other
QFT correlator and scattering amplitude calculations, e.g in MHV rules for
scattering amplitudes, as well as in ordinary QED. Their counting is a special
case of the counting of bi-partite embedded graphs. We review and extend
relevant mathematical literature and present results on the counting of some
infinite classes of bi-partite graphs. Permutation groups and representations
as well as double cosets and quotients of graphs are useful mathematical tools.
The counting results are refined according to data of physical relevance, such
as the structure of the vertices, faces and genus of the embedded graph. These
counting problems can be expressed in terms of observables in three-dimensional
topological field theory with S_d gauge group which gives them a topological
membrane interpretation.Comment: 57 pages, 12 figures; v2: Typos corrected; references adde
