Generalization of $1/N$ expansion in theories with matrix fields

The Feynman diagrams in theories with matrix fields $\Phi\in\mathop{\bf Mat}\nolimits(N,N)$ are ribbon graphs. There is a one-to-one correspondence between the ribbon graphs and surfaces with boundary having cell structures. The exponents of $N$ in the Feynman integrals depend only on the topology of the corresponding surfaces. It is shown that this is also true of generalized ribbon graphs. Such graphs correspond to surfaces with decomposition into spheres with holes.