On the refined counting of graphs on surfaces

Understand

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.