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Modulating the photon tunneling probability is the key to control the near-field radiative heat transfer (NFRHT) between two objects. It has been found that multiple body-vacuum interfaces can enable plentiful resonance modes which significantly amplify photon tunneling probability. In the present paper we demonstrate that a graphene/vacuum multilayers can support coupled multiple surface plasmon polaritons (MSPPs), not only enhancing but also suppressing the photon tunneling, hence the NFRHT. We demonstrate that, due to the plentiful symmetric and anti-symmetric branches produced by the coupling of MSPPs, enhancement of NFRHT is most prominent as the middle gap is comparable to the separation distance. Moreover, by adjusting the separation distances, number of layers and chemical potentials, several interesting phenomena are observed in the multilayers, hence providing more possibilities to …