To study the scattering properties of core-shell structure containing graphene, expecially electric field distribution, energy density, and radiative/nonradiative decay rate properties, the transfer matrix method (TMM) is used to evaluate the effects of wavelength of the incident ray and dipole emitter location on the spontaneous radiation of Au@SiO₂@Graphene. It is found that when the core size and shell thickness remain constant, the Purcell factor in the near-field region of the core-shell structure increases with the increase of the thickness of the intermediate SiO₂ dielectric layer. And the Purcell factor of the core-shell structure decreases with the distance of the dipole emitter from the center of the sphere, realizing the regulation of spontaneous radiation. On the other hand, by optimizing the structural parameters of graphene-based core-shell structure, the maximum value of the Purcell factor in the visible light range is achieved at about 633 nm. When the thickness of the dielectric layer is fixed, increasing the radius of the model will lead to a decrease of the Purcell factor. And changing the size parameters of the model can effectively strengthen the interaction between the graphene-based core-shell structure with light wave. The research can provide a reference for the solar thermal field.