Wiener index application in intuitionistic fuzzy rough graphs for transport network flow

The applicability of different topological indices is indispensable in fields such as chemistry, electronics, economics, business studies, medicine, and the social sciences. The most popular index in graph theory is the wiener index [Formula: see text], which is based on the geodesic distance between two vertices. It is assumed that the weight of the geodesic between vertex x and vertex y in intuitionistic fuzzy rough graphs (IFRG) is zero in the absence of a directed path. With regard to intuitionistic fuzzy rough graphs, the objective of this work is to investigate in detail the wiener index [Formula: see text] and the average wiener index ([Formula: see text]. Also, the connectivity index [Formula: see text] is one of the most significant indices, providing several examples and results. For intuitionistic fuzzy rough graphs, alternative distance and degree-based topological indices have also been developed. The research on intuitionistic fuzzy rough graphs that has been suggested is appropriate for representing imprecise data and uncertainty in practical situations. Additionally, examined is the connection between the wiener and connectivity indices. Finally, we proposed the use of wiener indices in transport network flow.