Carbon can form many allotropes. Because of the Carbon's valence and due to the atom arrangements in lattices, some of the allotropes can adopt different shapes and spatial conformations like planar sheets (graphene sheets), nanocones, nanotubes, nanobuds, nanoribbons, nanocages (pseudospheres or spheroids - fullerenes). Their extraordinary physical properties are promoting these structures as candidates with different applications in fields like Optoelectronics, Nanoelectronics, Biology (including Medicine and Pharmacy) and Electrochemistry. Because of their relatively new discovery, because of their complex molecular structures (presenting high numbers of possible isomers) and numerous possibilities of chemical transformations of these structures (addition reactions, substitution reactions, reactions through which two or more structures of this kind will covalently attach), their topology is still an object of great interest for scientists. Starting from the general aspects of connectivity in graphs, graph connectivity indices (Randic Index, Atom Bond Connectivity Index, Wiener Index, Balaban Index, Zagreb Indices, Hosoya Indices), new or adapted connectivity indices have been proposed for these structures to theoretically assess the atom arrangements. Because of their diverse dimensionality (Wiener dimensionality which refers to the atom connectivities and also geometrical one which refers to spatial conformations - planar or three dimensional structures), mathematical equations have been introduced to assess their atom bonding and topology (including volumes and surface areas). Computer software especially conceived for the calculus and design of these structures also exists today.