Topological Invariants of Nanocones and Fullerenes

Abstract

The Timisoara-eccentricity (TM-EC) index of a molecular graphis defined as the sum of  i i i over all at- oms i in � , where  i,  i and  i are the degree, eccentricity and the number of atoms at distance  i from atom i. The topo- logical efficiency index ofis defined as = 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 in- variants. It is proved that the TM-EC index of a fullerenecan 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 5n 2 + 10n + 5 carbon atoms, we have   1.24 and TM - EC = 280n 3 + 385n 2 + 195n + 40. Finally, we examine the dual of this nanocone and prove that we have   1.24 and TM - EC = 70n 3 + 20n 2 - 5n.