Gutman index of fuzzy graphs with application

Fuzzy graph theory is applicable on diverse fields where uncertainty plays a significant role like in network analysis, medical diagnosis and health care, decision-making, and optimization, etc. An effective tool in this field is the topological index. These indices provide quantitative measures of molecular structure and connectivity, which are crucial for understanding and predicting physical, chemical, and biological properties of molecules. The Gutman index (GI) is an important topological index. This index is based on degree and distance. In this paper, GI is explored in fuzzy graphs. Some results on GI are found like the relationship of GI between a fuzzy graph and its fuzzy sub-graph and the relationship of GI between a fuzzy graph and its complete fuzzy graph. Some upper and lower bounds are obtained for complete fuzzy graph, bipartite graph, star, double star, etc. A relationship of GI between two isomorphic fuzzy graphs are determined. Human trafficking, a heinous offense and grave breach of human rights involves the coercion, deception, or coercion of people for a range of purposes, including forced work, forced marriage, and forced servitude. Overall, human trafficking is a complex and multifaceted issue that requires coordinated efforts at the local, national, and international levels to prevent, prosecute, and ultimately eradicate this egregious violation of human dignity. At the end of this paper, an application of GI is given to reduce human trafficking in India.