  In this paper, we introduce a parameterized discrete curvature
($\alpha$-curvature) for piecewise linear metrics on polyhedral surfaces, which
is a generalization of the classical discrete curvature. A discrete
uniformization theorem is established for the parameterized discrete curvature,
which generalizes the discrete uniformization theorem obtained by
Gu-Luo-Sun-Wu. We also prove the global rigidity of parameterized discrete
curvature with respect to the discrete conformal factors, which confirms a
generalized Luo conjecture on rigidity of discrete curvatures. We further
introduce a parameterized discrete Yamabe flow for piecewise linear metrics on
surfaces. To handle the possible singularities along the flow, we do surgery on
the flow by flipping. Then we prove that the flow with surgery converges to a
piecewise linear metric with constant discrete $\alpha$-curvature, which
confirms another generalized Luo conjecture on convergence of discrete Yamabe
flow with surgery. We also introduce a parameterized discrete Calabi flow and
prove the convergence of the flow with surgery, which generalizes the
convergence proved by Zhu and the author.
