Wiener index of a directed rough fuzzy graph and application to human trafficking

In this paper, we introduced Wiener index (WI ) and average Wiener index (AWI ) of directed rough fuzzy graph (DRFG). WI is the most extensively used index in graph theory. This index is based on the geodesic distance between two vertices. If there is no directed path from vertex x to vertex y in DRFG, we assume that the weight of geodesic from vertex x to vertex y is zero. In this paper, we investigate the connection between WI and connectivity index (CI ), which is one of the most prominent index, by presenting several examples and results. We introduced the concept of complete directed rough fuzzy graph (CDRFG) along with some useful results like CDRFG have no weak edges. We also compute the WI for CDRFG. Moreover, we discussed three types of vertices: Wiener enhancing vertex (WEV), Wiener reducing vertex (WRV), and Wiener neutral vertex (WNV). The proposed study of DRFG is suitable for modeling uncertainties and unclear data information in the real life circumstances. In the end, we proposed an application of the WI in the human trafficking network. We also presented a detailed comparative analysis and comparison table by comparing our result for both CI and WI for the same human trafficking network.