Spectral energies of redefined Zagreb indices and comparative QSPR applications to anticancer and alcohol datasets

This article develops a unified framework for the redefined Zagreb descriptors that combines graph theory, spectral analysis, and molecular structure-property applications. We study the basic redefined Zagreb descriptors together with their higher-order variants, introduce the associated weighted graph matrices, and investigate their spectral radii, energies, and related structural interpretations. For several standard graph families, including paths, cycles, complete graphs, stars, complete bipartite graphs, wheels, and friendship graphs, we derive explicit formulas and show how these descriptors reflect different degree patterns and connectivity structures. We also establish general bounds for the descriptors and their weighted spectral quantities, thereby clarifying their connections with classical degree-based indices and adjacency energy. To examine chemical relevance, we first consider a small set of anticancer drug-like molecules and use it as an analytical descriptor study. In this setting, the redefined Zagreb descriptors and their energy-based analogues are strongly associated with size-related physicochemical quantities, while the mixed higher-order descriptor [Formula: see text] shows the strongest relationship with the minimum universal force-field energy. This part of the study is intended to identify informative descriptor trends rather than to establish a fully validated predictive model. We then carry out a broader QSPR study on 100 alcohol compounds with 17 physicochemical endpoints under repeated leakage-safe grouped external validation. The results show that the redefined Zagreb descriptor family and its derivative forms provide strong predictive performance for many targets, especially those related to molecular size, volume, and critical-property behavior. The derivative descriptors are therefore chemically meaningful and useful, while the combined representation shows where complementary information can be gained. Overall, the redefined Zagreb framework emerges as a mathematically rich and chemically useful family of descriptors whose combinatorial, spectral, and predictive roles can be studied in a unified way.