. A method for the structural classification of fullerenes via graph invariants is presented. These graph invariants (called edge-parameters) represent the 9 different types of bonds existing in fullerenes between two neighbouring carbon atoms and they are also applicable to classify the fullerene isomers into equivalence classes. Discriminating performance of edge-parameters has been tested on the sets of C40 and C66 fullerene isomers. It is shown that the stability of C40 and C66 isomers can be efficiently predicted using a novel topological descriptor (Ω) defined as a function of four appropriately selected edge parameters.