T-spherical fuzzy Zagreb index and energy in decision-making problems

The topological index, an essential tool in mathematical chemistry for associating network structures with numerical values, is now being utilized in T-spherical fuzzy graphs (T-SFGs), which extend both picture fuzzy graphs and spherical fuzzy graphs in graph theory. T-SFGs encompass three distinct opinions: truth membership ([Formula: see text]), abstain/neutral ([Formula: see text]), and falsity ([Formula: see text]), making them highly relevant for addressing real-world problems by capturing a wider range of perspectives. Compared to other fuzzy models, T-spherical fuzzy Zagreb index models deliver more efficient and accurate outcomes. This paper presents the definitions of the first and second Zagreb indices for T-SFGs, supported by illustrative examples for better understanding. It further introduces the corresponding first and second Zagreb matrices for T-SFGs, together with their energy and Laplacian energy measures. In addition, the study highlights the relationship between the directed graph associated with T-SFGs’ energy, Laplacian, and skew Laplacian energies for the Zagreb indices. Finally, the practical relevance of these concepts is demonstrated through their application in identifying the most suitable biological network for cancer treatment based on different protein types.