The semi-empirical Aboav-Weaire law is generally used to characterize the local topological correlation between the neighbor cells in 2D space-filling random cellular system. Starting with a possible generalization of the Aboav-Weaire law, this paper presents a new method, which is designated primarily to the topological characterization of finite cellular systems defined on an unbounded, closed and orientable surface, and composed of a finite set of combinatorial polygons. It is demonstrated that using two simple topological parameters (A(R) and E-R) derived from the extended Aboav-Weaire equation, the local structure and the stability of polyhedral fullerenes can be efficiently evaluated on the basis of quantitative criteria.