Connectivity index of directed rough fuzzy graphs and its application in traffic flow network

The directed rough fuzzy graph (DRFG) is a fusion of rough and fuzzy theory, as it deals with incomplete and vague information simultaneously. Connection or the strength of connectivity (SC) is vital in the realm of circuits or networks that are linked to the real world. As a result, SC is one of the most essential aspects of a directed rough fuzzy network system. The neighborhood connectivity index (NCI) is one such parameter that has a variety of applications in network theory. In this study, our main objective is to present a new topological index NCI based on DRFGs to solve complicated problems. Motivated by the modeling of networks, the strength of vertices to their neighboring vertices, and the efficiency of DRFGs to solve complex problems, we aim to study the NCI of DRFGs. In this paper, we successfully introduce a notion NCI based on DRFGs to deal with the uncertainties that arise in real-world problems. Based on the strength of vertices to neighboring vertices, we provide several lower and upper bounds for the NCI of DRFGs with reference to other graph invariants such as the number of vertices, edges, and degree distance. When we study NCI in operations for DRFGs with a large number of vertices, the degree of vertices in a DRFG provides a confusing picture. Therefore, a mechanism to determine the NCI for DRFG operations is therefore required. Therefore, generalized formulas for the NCI of DRFGs obtained by operations such as union, composition, and Cartesian product are also developed. An algorithm for obtaining the NCI of DRFGs is also proposed. Further, an application of the NCI of DRFGs in traffic flow networks was discussed to identify the busiest intersection using the proposed algorithm. Finally, we illustrate a comparative analysis and analysis table of the established approach (NCI) with existing techniques (connectivity index (CI) and Wiener index (WI)) are shown to demonstrate the validity of the presented approach.