Aqsa NazarFA23-RMT-005Prof. Dr. Muhammad HussainLHR TP 97532026-01-072025https://repository.cuilahore.edu.pk/handle/123456789/346This thesis investigates the structural and mathematical properties of the line graph of the HourGlass System Carbon Nano Cone using a wide range of topological indices derived from graph theory. Topological indices are numerical parameters that encapsulate the structural characteristics of a graph and have proven valuable in chemical graph theory for predicting the physicochemical properties of molecular compounds. The study focuses on two primary classes of indices: those based on vertex degree and those based on vertex neighborhood degree. A total of fifty vertex degree based and ninety vertex neighborhood degree-based indices are computed and analyzed for both even and odd cycle lengths of HGSCNC graphs. These include classical indices such as the Randic, Zagreb, and Atom-Bond Connectivity indices, as well as more recent and refined descriptors like the Generalized Somber Index, Hyper Zagreb Indices, Geometric-Arithmetic Index, and several Kulli–Basava variants. The HourGlass System Carbon Nano Cone is modeled as a finite, simple, undirected graph, satisfying all necessary conditions for the application of these indices. The results provide explicit analytical expressions and demonstrate the effectiveness of these indices in capturing the molecular complexity and connectivity of the HGSCNC structure. This work not only broadens the understanding of graph-theoretical modeling in nanostructures but also reinforces the applicability of topological indices in materials science, nanotechnology, and quantitative structure property relationship (QSPR) studies.enProf. Dr. Muhammad HussainMATHEMATICSDifferent Chemical NetworkThesis