We have designed and proposed the edge modes supported by graphene ribbons and the planar band-pass filter consist-ing of graphene ribbons coupled to a graphene ring resonator by using the finite-difference time-domain numerical method. Simulation results show that the edge modes improve the electromagnetic coupling between devices. This structure works as a novel, tunable mid-infrared band-pass filter. Our studies will benefit the fabrication of planar, ultra-compact nano-scale devices in the mid-infrared region. A power splitter consisting of two output ribbons that is useful in photonic integrated devices and circuits is also designed and simulated. These devices are useful for designing ultra-compact planar devices in photonic integrated circuits.