Logic gates based on optical waveguides have been implemented in recent years. But the use of the plasmonic effect on logical devices like semiconductor has not yet been investigated. The graphene-based plasmonic effect is a novel effect that is discussed in this paper on nanostructure devices such as semiconductor. In this paper, a logical innovation for the design of the logical gates AND and XOR has been implemented as the basic components for the half adder circuit based on graphene plasmonic waveguides. The functions for the graphene are computed by the equations and charts defined in the articles and applied to the system. By applying the environmental parameters, the conductivity value at the chemical potential of 0.565 ev is obtained according to the conductivity equation of 0.936 × 10-4. Due to the constant value of the permeability of the vacuum, which is equal to one and the graphene thickness is equal to 0.7 nm, the amount of the dynamic part of the dielectric coefficient at the 7 Grad / s work frequency is the approximate value of 0.02. The wavelength of the input signal is 1.55 μm in the chemical potential of 0.565 ev. After applying the defined parameters of graphene under the plasmonic waveguide for the desired structure, there are good results for responding to the logical structures in half adder circuit. The first designed device is a graphene-based AND plasmon structure. The power factor of this switch is 0.98 for optimum on state and 0.55 for off state. Then the structure of XOR is designed, whose power factor of this switch in optimal conditions is 0.85 for on state and 0.02 for off mode, which a in comparison with previous works, have maximum power transfer characteristics in the half adder circuit.