A negative value for the nonlinear refraction in graphene is experimentally observed and unambiguously verified by performing a theoretical analysis arising from the conductivity of the graphene monolayer. The nonlinear optical properties of multi-layer graphene are experimentally studied by employing the Z-scan technique. The measurements are carried out at 1150, 1550, 1900 and 2400 nm with a 100-femtosecond laser source. Under laser illumination the multi-layer graphene exhibits a transmittance increase due to saturable absorption, followed by optical limiting due to two-photon absorption. The saturation irradiance I-sat and the two-photon absorption coefficient beta are measured in the operating wavelength range. Furthermore, an irradiance-dependent nonlinear refraction is observed and discriminated from the conventional nonlinear refraction coefficient n(2), which is not irradiance dependent. The values obtained for the irradiance-dependent nonlinear refraction are in the order of similar to 10(-9) cm(2)W(-1), approximately 8 orders of magnitude larger than any bulk dielectrics.