The structural properties of sulfur superstructures adsorbed on Ru(0001) have been widely studied in the past. However, much less effort has been devoted to determine their electronic properties. To understand the connection between structural and electronic properties, we have carried out density functional theory periodic boundary calculations mimicking the four long-range ordered sulfur superstructures identified experimentally by means of scanning tunneling microscopy (STM) techniques. Our simulations allow us to characterize the nature of the sulfur-Ru bond, the charge transfer between the Ru substrate and the sulfur adlayers, the interface states, and a parabolic state recently identified in STM experiments. A simple analysis, based on a one-dimensional model, reveals that this parabolic state is related to a potential well state, formed in the surface when the concentration of sulfur atoms is large enough to generate a new minimum in the surface potential.