Quantitative structure-activity and structure-property associations of natural systems require terminologies for their topological properties. Structure-based topological descriptors/indices of these systems allow these chemical possessions and the bioactivities of these compounds through reckonable structure-activity and structure-property associations’ procedures. In this paper, we propose a computational technique to compute analytically exact expressions for certain degree and distance-based topological indices for general graphs. A comparative analysis is conducted with the known techniques where certain experiments are performed to show that our technique is more efficient and possesses less algorithmic and computational complexity. We apply our method to compute explicit expressions of certain degree and distance topological indices for certain infinite families of fullerenes, carbon nanotubes, and carbon nanocones. The obtained results in this paper generalize certain known results in the literature.