The zero-voltage conductance of nano-graphenes: Simple rules and quantitative estimates

Zero-dimensional graphenes, also called nano-graphenes ($NGs$), consist of fragments of graphene with a finite number of hexagons. $NGs $ can be viewed as a subset of the polycyclic aromatic hydrocarbons (PAHs) that continue to attract chemists attention. We develop a simple theory for the ballistic electron transport through $NGs$ which combines elements of the electronic structure theory of graphene, intuitive methods for the calculation of the molecular conductance, and chemical concepts such as Kekul\'{e} structures. This theory enables one to analyze the relation between the structure of $NGs$ and their conductance. General formulas and rules for the zero-voltage conductance as a function of the contact positions are derived. These formulas and rules require at most simple paper and pencil calculations in applications to systems containing several tens of carbon atoms.