  We derive an inequality that relates nodal set and eigenvalues of a class of
twisted Dirac operators on closed surfaces and point out how this inequality
naturally arises as an eigenvalue estimate for the $\rm Spin^c$ Dirac operator.
This allows us to obtain eigenvalue estimates for the twisted Dirac operator
appearing in the context of Dirac-harmonic maps and their extensions, from
which we also obtain several Liouville type results.
