  The strong-coupling Eliashberg theory plus vertex correction is used to
calculate the maps of transition temperature (Tc) in parameter-space
characterizing superconductivity. Based on these Tc maps, complex crossover
behaviors are found when electron-phonon interaction increases from
weak-coupling region to strong-coupling region. The doping-dependent Tc of
cuprate superconductors and most importantly the pseudo-gap can be explained as
the effects of vertex correction.
