Localized surface plasmonic resonance has attracted extensive attention since it allows for great enhancement of local field intensity on the nanoparticle surface. In this paper, we make a systematic study on the excitation of localized surface plasmons of a graphene coated dielectric particle. Theoretical results show that both the intensity and frequency of the plasmonic resonant peak can be tuned effectively through modifying the graphene layer. Furthermore, high order localized surface plasmons could be excited and tuned selectively by the Laguerre Gaussian beam, which is induced by the optical angular orbital momentum transfer through the mutual interaction between the particle and the helical wavefront. Moreover, the profiles of the multipolar localized surface plasmons are illustrated in detail. The study provides rich potential applications in the plasmonic devices and the wavefront engineering nano-optics.