The conductive and mechanical properties of polymer composites based on styrene-butadiene rubber with different concentrations of graphene as a filler are studied. Based on the dependence of the resistivity of composites on the graphene content, the range of its values corresponding to the hopping conductivity mechanism was determined. For composites of compositions from this range, the sensitivity of the resistance to the degree of uniaxial stretching, as well as mechanical properties were investigated. The values of the gauge factor were determined, and it was found that with an increase in graphene content in the composite, its value first increases and then decreases. It is found that even under minimal stresses in composites, along with elastic deformations, plastic deformations also develop, and the proportion of the latter increases with the filler concentration. As a result, complete mechanical relaxation of the composites does not occur after the applied stress is removed.