Unveiling the Interplay Between Degree-Based Graph Invariants of a Graph and Its Random Subgraphs

This paper investigates the significance of employing random subgraphs and analyzing the expected values of Zagreb ndices within chemical graphs. By examining smaller, representative subsets, we uncover valuable insights into the properties and characteristics of complex networks. The expected values of Zagreb indices serve as critical mathematical measures for quantifying the structural complexity of chemical graphs, providing essential information about connectivity and branching patterns within molecules. Our primary contribution includes deriving theoretical expressions for these indices and validating them through extensive computational experiments on fullerene graphs $C_{20}$ $C_{20}$ and $C_{60}$ $C_{60}$ . The results demonstrate that our theoretical predictions closely align with experimental findings, affirming the robustness of Zagreb indices in characterizing molecular structures. Additionally, our analysis of specific cases, such as complete graphs and complete bipartite graphs, is consistent with previous studies, further reinforcing our methodology. This research emphasizes the relevance of random subgraphs and expected values of Zagreb indices in advancing our understanding of molecular behavior and stability, with important implications for materials science and drug design.