The Timisoara-eccentricity (TM-EC) index of a molecular graph is defined as the sum of δiεiζi over all
atoms i in Γ, where ζi, εi and ζi are the degree, eccentricity and the number of atoms at distance εi from atom i. The
topological efficiency index of ζ is defined as ρ = 2W / Nw , where W denotes the Wiener index, w is the minimal
vertex contribution and N is the number of carbon atoms. This paper is devoted to the study of nanocones and
fullerenes by these new graph invariants. It is proved that the TM-EC index of a fullerene ζ can be bounded by a
polynomial of degree 2, for twelve infinite series of fullerenes. It is also shown that in one pentagonal carbon nanocone with exactly 5n2
+ 10n + 5 carbon atoms, we have ρ ≈ 1.24 and TM - EC = 280n>sup>3 + 385n2 + 195n + 40. Finally, we examine the dual of this nanocone
and prove that we have ρ ≈ 1.24 and TM - EC = 70n3 + 20n2 - 5n.