Atomic force microscopy (AFM) measurements show that the Ag nanoparticles that yield surface-enhanced Raman scattering (SERS) of single molecules of Rhodamine (R6G) are all compact aggregates consisting of a minimum of two individual particles. Comparison of 514.5 and 632.8 nm excitation shows that the single molecule R6G signal is significantly higher when the excitation wavelength is resonant with the absorption band of R6G and suggests that the Raman excitation spectrum follows the absorption profile for R6G. We have also observed an interesting superlinear power dependence of the SERS signal. On average, by increasing the incident power by 2 orders of magnitude and decreasing the integration time by the same factor to maintain constant fluence, increases of 4 to 6 times were observed in the SERS intensity. We discuss these results in terms of model where the R6G molecule that yields single molecule SERS signals is located at the junction of two Ag nanoparticles. We have also modeled the system using molecular resonance Raman theory to provide further insight into the enhancement mechanism.