Graphene is used as an ideal platform for plasmonic optical devices due to its unique electrical and optical characteristics. We proposed a unique effective index method to evaluate a graphene-based half Maxwell fish-eye (HMFE) plasmonic lens on a single flake of graphene, which can focus a surface plasmon polariton plane wave and transform into a spherical wave. Similar to traditional optics, the proposed method can be used to investigate the focusing performance of an HMFE lens versus different parameters, including the incident frequencies, the discrete semiring numbers, the chemical potentials, and the size of the proposed lens. In addition, we use two proposed lenses to build an optical-coupling element, which can transform the beam of a plane wave into a different beam width. The proposed effective index method can be used as a reference in designing plasmonic optical devices with a variable effective index profile on a flake of graphene. The finite element method is employed to realize the numerical simulation, which demonstrates results almost consistent with our design methodology.