Multiplicative Version of First Zagreb Index in Fuzzy Graph and its Application in Crime Analysis

The multiplicative first Zagreb index is discussed in this paper and studied for fuzzy graphs. Bounds of this index are calculated for path, cycle, star, complete fuzzy graph, partial fuzzy subgraph, etc. For an isomorphic fuzzy graph, it is shown that the value of this index is the same. Some interesting relations are established among the first Zagreb index, F-index and multiplicative first Zagreb index for a fuzzy graph. The bounds of this index are studied for some fuzzy graph operations. Also, the crime of kidnapping and abduction in metropolitan cities in India is analyzed by this index. Significant Statement: Topological indices for a crisp graph have applications in real life. But sometimes, it is seen that most real-life problems cannot be modeled using crisp graphs. Also, laboratory testing of chemicals to understand their different properties are costly. To overcome this, many topological indices have been presented in theoretical chemistry. Those topological indices are needed to define for a fuzzy graph for these circumstances.