  We present a method to compute the genus expansion of the free energy of
Hermitian matrix models from the large N expansion of the recurrence
coefficients of the associated family of orthogonal polynomials. The method is
based on the Bleher-Its deformation of the model, on its associated integral
representation of the free energy, and on a method for solving the string
equation which uses the resolvent of the Lax operator of the underlying Toda
hierarchy. As a byproduct we obtain an efficient algorithm to compute
generating functions for the enumeration of labeled k-maps which does not
require the explicit expressions of the coefficients of the topological
expansion. Finally we discuss the regularization of singular one-cut models
within this approach.
