We explore the stability of different galaxy light concentration indices as a function of the outermost observed galaxy radius. With a series of analytical light-profile models, we show mathematically how varying the radial extent to which one measures a galaxy's light can strongly affect the derived galaxy concentration. The ``mean concentration index,'' often used for parameterizing high-redshift galaxies, is shown to be horribly unstable, even when modeling one-component systems such as elliptical, dwarf elliptical, and pure exponential disk galaxies. The C31 concentration index performs considerably better but is also heavily dependent on the radial extent, and hence exposure depth, of any given galaxy. We show that the recently defined central concentration index is remarkably stable against changes to the outer radius and observational errors and provides a both meaningful and reliable estimate of galaxy concentration. The Sérsic index n from the r1/n models is shown to be monotonically related with the central concentration of light, giving the index n a second and perhaps more tangible meaning. With a sample of elliptical and dwarf elliptical galaxies, we present correlations between the central light concentration and the global parameters: luminosity (Pearson's r=-0.82), effective radius (r=0.67), central surface brightness (r=-0.88), and velocity dispersion (r=0.80). The more massive elliptical galaxies are shown to be more centrally concentrated. We speculate that the physical mechanism behind the recently observed correlation between the central velocity dispersion (mass) of a galaxy and the mass of its central supermassive black hole may be connected with the central galaxy concentration. That is, we hypothesize that it may not simply be the amount of mass in a galaxy but rather how that mass is distributed that controls the mass of the central black hole.