  The Purcell factor and the spontaneous emission spectrum of a quantum emitter
(QE) placed close to the edge of a graphene half-space nanostructure is
investigated, using semi-analytical methods at the electrostatic regime. The
half-space geometry supports an edge and a bulk surface plasmon (SP) mode. The
Purcell factor of the QE is enhanced over eight orders of magnitude when its
emission energy matches the resonance energy modes, for a specific value of the
in-plane wave vector, at a separation distance of $5\,$nm. The different
transition dipole moment orientations influence differently the enhancement
factor of a QE, leading to large anisotropic behavior when positioned at
different places above the half-space geometry. The field distribution is
presented, showing clearly the excitation of the SP modes at the edge of the
nanostructures. Also, we present the spontaneous emission spectrum of the QE
near the half-space graphene nanostructure and show that strong light-matter
coupling may emerge. When a QE with a free-space lifetime of $1\,$ns is placed
at a distance of $10\,$nm away from the edge of the graphene half-space, a Rabi
splitting of $80 \,m$eV is found. Our contribution can be used for designing
future quantum applications using combination of QEs and graphene
nanostructures.
