We introduced 4 types of topological features, namely, atom, bond, pentagon, and hexagon features to characterize the structure of fullerenes. Each carbon atom in the fullerene cage can form three C-C bonds, and each bond is surrounded by four polygonal rings which belong to either pentagon or hexagon. The atom features categorize carbon atoms into 4 types, denoted as {n i |i = 0, 1, 2, 3}, representing the count of carbon atoms fused to i adjacent pentagons (Figure ). Similarly, the bond features define 9 types of C-C bonds {e i |i = 0, . . . , 8} according to the number and arrangements of the 4 surrounding polygonal rings (Figure ). The pentagon features define 8 types of pentagons {p i |i = 0, . . . , 7} depending on the number of arrangements of the 5 abutting polygons (Figure ). The hexagon features define 13 types of hexagons {h i |i = 0, . . . , 12} which rely on the number and arrangements of the 6 neighboring polygons (Figure ). Table lists the values of these 4 topological features for C 20 -#1, C 60 -#1, and C 60 -#1812. It is important to note that Fowler and Manolopoulos also designed pentagon {p ′ i |i = 0, . . . , 5} and hexagon {h ′ i |i = 0, . . . , 6} indices; however, they only considered the number of pentagons (hexagons) attached to i other pentagons (hexagons) but ignored different arrangements of the pentagons (hexagons). For instance, p ′ 2 equals the sum of p 2 and p 3 , whereas p ′ 3 equals the sum of p 4 and p 5 (see Figure ). They further consolidated the pentagon and hexagon indices into a more practical set of topological indices, namely, the pentagon signature P 1 , and the n-th moment hexagon signature H n using the formulas below: