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Electromagnetic waves with orbital angular momentum (OAM) have garnered significant interest by reason of their potential in designing advanced communication systems. This paper reports on a specific design of a flexible graphene-based antenna for RF vortex wave generation. The antenna is composed of a ring-shaped resonator made of a graphene film with high conductivity deposited on a thin polydimethylsiloxane substrate. The graphene film is characterized by greater strength to repeated mechanical bending than metallic counterparts while providing good conductivity of the patch. We use the commercial ANSYS HFSS 3D electromagnetic solver to conduct full-wave numerical simulations of the antenna performances and radiation conditions, simulating the generation of vortex waves at different bending radii of the antenna. The mode purity is evaluated with the use of the spiral spectrum algorithm to reveal the stability of the vortex wave generation by the antenna in bent states. Several pilot samples of the antenna working in the RF band are fabricated and tested. Calculated together with empirical results demonstrate that the flexible antenna can stably generate vortex waves even subjected to either outward or inward bending. The proposed flexible graphene-based antenna has the potential for application in wireless communication systems.