A generalized entropy framework for topological indices on specific graphs

This paper introduces a generalized entropy framework for analyzing the structural complexity of graph families through degree-based topological indices. We define a unified parametric entropy functional that encompasses Shannon entropy derived from multiple topological indices—including Zagreb indices (first and second), Forgotten index, Augmented Zagreb index, Sombor index, Elliptic Sombor index, Randić index, and Nirmala index—applied to three graph classes: Helm graphs ( Hₙ ), Fan graphs ( F1,n ), and Friendship graphs ( D₃ⁿ ). Beyond explicit entropy computations, this work makes the following theoretical contributions: (i) we establish rigorous asymptotic bounds proving that all index-based entropies converge as n → ∞ with explicit convergence rate O(log n / n) ; (ii) we prove monotonicity theorems guaranteeing that entropy is strictly increasing with graph order n for each index; (iii) we introduce a novel Hybrid Structural Entropy (HSE) index that aggregates information from multiple topological indices via a weighted convex combination, and prove it achieves tighter discrimination between graph families than any individual index-based entropy. To address the absence of applied validation in prior work, we conduct QSPR regression analysis correlating our entropy measures with experimentally determined boiling points of alkane isomers (C 5 –C 9 ), achieving R² values exceeding 0.94. Statistical validation includes Pearson and Spearman correlation tests with p -values, regression diagnostics, and comparative benchmarking against existing entropy models. We further extend the analysis to spectral entropy (von Neumann entropy of the normalized Laplacian) and compare it with degree-based entropies. The computational framework is fully reproducible, with all derivations performed symbolically in MATLAB.