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The thermoelectric effect in epitaxial graphene formed on the surface of a size-quantized metal film is examined in the context of a Davydov model. An approach based on the Kubo formula for the conductivity and differential thermal emf is used. It is shown that because of size quantization, near the edges of the energy levels the thermal emf of epitaxial graphene increases to gigantic values of 200 μV/K, or almost a factor of seven greater than the emf of isolated graphene (about 30 μV/K).