Bounds on F-Index of Tricyclic Graphs with AI Applications

Abstract

Topological indices, symmetric functions in graph theory, are critical tools for characterizing the structural and topological properties of molecules and networks. The Forgotten index (F-index), defined as the sum of the cubes of vertex degrees in a molecular graph, was intro duced by Furtula and Gutman to model the structural dependence of total π-electron energy. Tricyclic graphs, connected graphs of order m and size m+2, are of particular interest due to their intermediate complexity between bicyclic and tetracyclic structures. In this work, we establish sharp upper and lower bounds for the Forgotten index within the class of tri cyclic graphs and identify the extremal graphs achieving these bounds. To extend the theoretical contributions, we also propose to integrate artificial intelligence (AI) and machine learning (ML) methodologies, for their utility in three key areas: (1) Predictive modeling via graph neural networks (GNNs) to estimate F-indices for large-scale tricyclic graphs, bypassing combinatorial complexity; (2) Generative design using variational autoencoders (VAEs) to synthesize novel tricyclic graphs with near-optimal F indices for materials science applications; and (3) AI-driven optimization employing reinforcement learning (RL) to validate extremal graph structures and explore uncharted regions of the tricyclic graph space. We combine mathematical and AI approaches to improve F-index analysis in tricyclic graphs.