  We consider the large $N$ expansion of the partition function for the
Hermitian one-matrix model. It is well known that the coefficients of this
expansion are generating functions $F^{(g)}$ for a certain kind of graph
embedded in a Riemann surface. Other authors have made a simplifying assumption
that the potential $V$ is an even function. We present a method for computing
$F^{(g)}$ in the case that $V$ is not an even function. Our method is based on
the string equations, and yields "valence independent" formulas which do not
depend explicitly on the potential. We introduce a family of differential
operators, the "string polynomials", which make clear the valence independent
nature of the string equations.
