  In this paper, we prove various results for circle actions on compact unitary
manifolds with discrete fixed point sets, generalizing results for almost
complex manifolds. For a circle action on a compact unitary manifold with a
discrete fixed point set, we prove relationships between the weights at the
fixed points. As a consequence, we show that there is a multigraph that encodes
the fixed point data (a collection of multisets of weights at the fixed points)
of the manifold; this can be used to study unitary $S^1$-manifolds in terms of
multigraphs. We derive results regarding the first equivariant Chern class,
obtaining a lower bound on the number of fixed points under an assumption on a
manifold. We determine the Hirzebruch $\chi_y$-genus of a compact unitary
manifold admitting a semi-free $S^1$-action, and obtain a lower bound on the
number of fixed points.
