In this article, using energy method, the effect of surface stress on the free vibration and bending analysis of single-layer graphene sheet embedded in an elastic medium based on nonlocal elasticity theory is studied. Surface stress plays a more important role due to the high surface-to-volume ratio in nanoscale materials. For this purpose, Gurtin-Murdoch continuum mechanics approach is used. The effects of surface properties including surface elasticity, surface residual stresses and surface mass density are considered. In this research, the effects of Winkler spring constant, Pasternak shear constant, aspect ratio, Young's modulus of surface layer, surface residual stress, nonlocal parameter on the natural frequency ratio and deflection of single-layer graphene sheet are investigated. The results indicate that the natural frequency ratio decreases with an increase in the surface residual stress and vice versa for deflection of nonlocal single-layer graphene sheet. Also, the natural frequency ratio increases with an increase in Young's modulus of surface layer and vice versa for deflection of nonlocal single-layer graphene sheet. Moreover, the elastic medium causes to stiffen the single-layer graphene sheet. The influence of the small-scale parameter on the natural frequency ratio and deflection of single-layer graphene sheet is more significant for lower values of foundation parameters and vice versa for higher values of foundation parameters.