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This article illustrates a design and finite difference time domain (FDTD) method based on analysis of fiber optic surface plasmon resonance (SPR) biosensor for biomedical application especially for DNA-DNA hybridization. The fiber cladding at the middle portion is constructed with the proposed hybrid of gold (Au), graphene, and a sensing medium. This sensor can be recognized adsorption of DNA biomolecules onto sensing medium of PBS saline using attenuated total reflection (ATR) technique. The refractive index (RI) is varied owing to the adsorption of different concentration of biomolecules. Result states that the sensitivity with a monolayer of graphene will be improved up to 40% than bare graphene layer. Owing to increased adsorption capability of DNA molecules on graphene, sensitivity increases compared to the conventional gold thin film SPR biosensor. Numerical analysis shows that the variation of the SPR angle for mismatched DNA strands is quite negligible, whereas that for complementary DNA strands is considerable, which is essential for proper detection of DNA hybridization. Finally, the effect of Electric field distribution on inserting graphene layer is analyzed incorporating the FDTD technique by using Lumerical FDTD solution software.