Multi-attribute decision-making using q-rung orthopair fuzzy Zagreb index

The q-rung orthopair fuzzy set (q-ROFS), an extension of intuitionistic and Pythagorean fuzzy sets, offers greater flexibility in representing vague information with two possible outcomes, yes or no. The fuzzy Zagreb index is an important graph parameter, widely used in fields such as network theory, spectral graph theory, mathematics, and molecular chemistry. In this paper, the first and second Zagreb indices for q-rung orthopair fuzzy graphs (q-ROFGs) are introduced, and bounds for these indices are established, including their behavior in regular q-ROFGs. Additionally, it is explored, how various graph operations such as union, Cartesian product, direct product, and lexicographical product affect the first Zagreb index. Furthermore, a new approach is presented that combines Multiple-Attribute Decision-Making (MADM) with graph-based models to improve decision-making, particularly in vaccine selection. The methodology constructs a bipartite graph for each attribute, where virologists assign membership and non-membership values to vaccines. The Zagreb index is used to measure the importance of each vaccine, and a weighted aggregation technique normalizes the scores. The final ranking is derived from a computed score function. The results demonstrate the effectiveness of the approach in providing a systematic and mathematically rigorous framework for multi-attribute decision-making, with rank correlation analysis confirming its robustness compared to existing methods such as q-ROF PROMETHEE, q-ROF VICOR, q-ROF TOPSIS, q-ROFWG, and q-ROFWA.