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In this work, the tuning property and sensitivity enhancement of the long period fiber grating (LPFG) coated with the higher refractive index film are demonstrated theoretically by integrating a monolayer graphene. The general variation rule of the mode characteristics and the resonance with the chemical potential of the graphene are explored. The polarization-independent transmission and sensing characteristics are obtained even though the optical property of the graphene is polarization-dependent. The results reveal that the sensing characteristics are significantly dependent on the tunable state of the graphene, which can be used to greatly optimize the sensing performance. By this approach, an ultrahigh sensitivity up to 28 337.5 nm/RIU is achieved, which is 2.57 times higher than that of the conventional LPFG without graphene. The sensitivity can be further optimized by integrating few-layer graphene. This tunable property at a wide range makes the graphene integrated LPFG devices without destroying the fiber integrity ideal for wide applications, such as biochemical sensing and optical modulation. [ABSTRACT FROM AUTHOR]