Application of Symmetric Division Deg Index of Graph Under Fuzzy Environment

Fuzzy graph theory, renowned for its capacity to manage uncertainty and capture varying degrees of relationships, has surfaced as a powerful framework for modeling and addressing complex problems across diverse domains such as medicine, social networks, and biological systems. Among the most effective analytical tools within this framework are topological indices, which play a crucial role in characterizing structural features and predicting the properties of chemical compounds and other network-based systems. In this study, we introduce the Symmetric division deg index (SDD) for fuzzy graphs (FGs), a degree-based topological index with potential for broader applications. We compute the SDD index values for regular FG, fuzzy cycle, fuzzy star, broom graph, and complete bipartite FG. Additionally, we determine extremal SDD index values for fuzzy tree, complete bipartite FG, and molecular FGs. The relation between two isomorphic fuzzy graphs are found. Furthermore, we define the average SDD index for fuzzy graphs and explore its connection with the SDD index for specific classes such as fuzzy cycles, complete FGs, and fuzzy stars. To facilitate computation, we present an algorithm and provide a Python implementation for calculating the SDD index of any fuzzy graph. At last, an application is given to reduce COVID-19 transmission.