A novel and integrated plasmonic full adder based on cascaded rectangular ring resonators for optical computing is presented. This full adder consists of cascaded rectangular ring resonators covered by graphene layer and resonator-enhanced waveguide crossings. Owing to the nonlinear optical properties of graphene, the plasmonic resonators can work as all-optical switches with low power consumption and ultrashort response time to change the propagation path of surface plasmon polaritons (SPPs). Finite-difference time-domain (FDTD) simulation results numerically verify the realization of full-adder operations. This proposed plasmonic device can realize the optical full-adder operations with small feature size, fast response time, and lower power consumption than when using other traditional nonlinear optical Kerr materials. It can provide new concepts to design novel optical computing devices.