Considering the anharmonic vibrations and the interactions between electron and phonon of atoms, in this article we study the temperature dependence of Grneisen parameter, thermal expansion coefficient at low temperature and phonon relaxation time by using the theory and method of solid state physics. The influences of the anharmonic vibration of the atom on the above parameters are further discussed. The obtained results are as follows. 1) The thermal expansion coefficient of graphene is a negative value when the temperature drops below room temperature. The absolute value of the thermal expansion coefficient of graphene increases monotonically with the increase of temperature. The thermal expansion coefficient of graphene is-3.64×10-6 K-1 at room temperature. 2) The value of Grneisen parameter is zero in the harmonic approximation. If the anharmonic vibration is considered, the Grneisen parameter will increase slowly with the increase of temperature. Its value is between 1.40 and1.42 and the change is almost linear. And we find that the influence of the second anharmonic term is less than that of the first anharmonic term on Grneisen parameter. 3) The phonon relaxation time decreases with the increase of temperature. The rate changes rapidly at low temperature (T<10 K), then it changes very slowly. The phonon relaxation time is almost inversely proportional to temperature when the temperature is higher than 300 K.