Anisotropic triangular graphene monolayers grown on a Ru(0001) surface represent unique two-dimensional templates for creating ordered, large-scale assembly of functional molecules. Using scanning tunneling microscopy and spectroscopy, we demonstrate the selective adsorption and the formation of ordered molecular arrays of iron phthalocyanine and pentacene molecules of different structural symmetries on the graphene/Ru(0001) templates. With in-depth investigations of the molecular adsorption and assembly processes, we reveal the existence of site-specific, lateral electric dipoles (or lateral electric fields) in the epitaxial graphene monolayers and the capability of the dipoles in directing and driving the molecular adsorption and assembly. We show that the lateral dipoles originate from the inhomogeneous distribution of charge due to the epitaxial constraint of graphene on a Ru(0001) surface. The adsorption mechanism is rather general and applicable to similar molecular systems on graphene monolayers formed on other transition metal surfaces. © 2011 American Physical Society.