Properties of the forgotten index in bipolar fuzzy graphs and applications.

Topological indices are the numbers that remain constant under graph automorphism. Topological indices describe a network's connectivity, structure, and topological characteristics. These indices have many applications in crisp graphs. However, in many cases, it is observed that some situations can't be described using the idea of crisp graphs. So, to overcome this issue, the need to define topological indices for fuzzy and bipolar fuzzy graphs arises. The F-index, or the Forgotten Index, is a significant topological index. A bipolar fuzzy graph with two opposite-sided opinions of both the edges and vertices measures the impreciseness or uncertainties of the edges and vertices along the positive and negative sides. In this article, we have presented the Forgotten Index for bipolar fuzzy graphs. Then, we have proved some theorems regarding the F-index of numerous types of bipolar fuzzy graphs, such as regular bipolar fuzzy graphs, complete bipolar fuzzy graphs, etc., the bounds of the F-index in bipolar fuzzy graphs, and the relationships of the F-index with other topological indices in bipolar fuzzy graphs. We have applied the proposed topological index, the F-index for bipolar fuzzy graphs, to matrimonial websites to find potential life partners based on compatibility and discussed the application of the Forgotten Index in gene regulatory networks.