First Entire Zagreb Index of Fuzzy Graph and Its Application

The first entire Zagreb index (FEZI) is a graph parameter that has proven to be essential in various real-life scenarios, such as networking businesses and traffic management on roads. In this research paper, the FEZI was explored for variety of fuzzy graphs, including star, firefly graph, cycle, path, subgraph, vertex elimination, edge elimination. This study presented several results, determining relationship between two isomorphic graphs path cycle (connecting both end vertices path). also deals with analysis α-cut establishes bounds some graphs. To apply these findings modern life problems, team utilized results identify areas require more development internet systems. These have practical implications enhancing efficiency effectiveness conclusion drawn from can used inform future aid efficient effective systems fields.