On Some Bounds of Clique Sombor Index and the Effect of Edge or Vertex Removal in a Graph

In this paper, we investigate the Clique Sombor Index (CS), a clique-based degree topological index. We establish upper and lower bounds for the Clique Sombor Index of general graphs and identify classes of graphs for which these bounds are attained. A comparative study with the well-known Atom–Bond Connectivity (ABC) index is carried out to highlight the similarities, differences, and the sensitivity of both indices to structural variations in graphs. Moreover, the impact of removing edges and vertices on the Clique Sombor Index is assessed, showing the impact of localized modifications of the graph structure on the index.This study shows that the Clique Sombor Index captures important clique-based structural properties of a graph and highlights the use of the index in lieu of classical degree-based topological indices in chemical graph theory and the analysis of complex networks.