ABSTRACT
Topological efficiency index is provided as a measure of the overall topological stability of small fullerenes; since then it has been applied to investigate stable configurations of large fullerenes and other organic compounds with n carbon atoms including infinite graphenic layers. Since the late 80’s the stability of Cn fullerene molecules is a complex puzzle for theoreticians that attack this target by using symmetry and structural considerations as well as sophisticated ab-initio quantum mechanical simulations. Innovative applications of topological efficiency invariants are already described in Wiener index, including the recent applications to neuroscience the most promising sector in which topological efficiency indices will find useful correlations between system topology and neural functions. Like the Wiener index, topological efficiency invariants represent optimal tools for investigating large nanostructures. Topological methods express the role of long-range interactions and are therefore capable of describing new kinds of phase-transition governed by these interactions.
