The phonon density of states (PDOS) and the thermodynamical properties including the heat capacity,the free energy,and the entropy of a single-layer graphene with vacancy defects have been studied theoretically.We first analytically derive the general formula of the lattice vibration frequency,and then numerically discuss the effect of the defects on the PDOS.Our results suggest that the vacancy defects will induce the sawtooth-like oscillation of the PDOS and the specific oscillation pattes depend on the concentration and the spatial distribution of the vacancies.In addition,it is verified that the vacancy defects will cause the increase of the heat capacity because of the vacancy-induced low-frequency resonant peak.Moreover,the influences of the vacancies on the free energy and the entropy are investigated.