  To distinguish the contributions to the generalized Hurwitz number of the
source Riemann surface with different genus, we define the genus expanded
cut-and-join operators by observing carefully the symplectic surgery and the
gluing formulas of the relative GW-invariants. As an application, we get some
differential equations for the generating functions of the generalized Hurwitz
numbers for the source Riemann surface with different genus, thus we can
express the generating functions in terms of the genus expanded cut-and-join
operators.
