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A Survey on Wiener Index in Fuzzy Graphs with Application in Medical Screenings
Ruiqi Cai, Maryam Akhoundi, Janusz Kacprzyk, Aysha Khan, Jana Shafi and Hossein Rashmanlou
A vague graph (VG) is one of the versatile application tools in the field of mathematics, which allows the user to easily describe the vague relation between any objects. VGs are beneficial to give more precision and flexibility to the system as compared to the classical model (i.e.,) crisp theory. The topological index (TI) of graph has a wide range of applications in theoretical chemistry, network design, computer science, etc. In chemical graph theory, the wiener index (also wiener number) introduced by HarryWiener, is a topological index of a molecular, defined as the sum of the lengths of the shortest paths between all pairs of nodes in the chemical graph. Many topological indices exist only in the crisp but it’s new to the VG-environment. The main aim of this research work is to define the topological indices in VGs. Here, indices described in VGs are Hyper Wiener index (HWI), Zagreb index (ZI), Randic index (RI), Harmonic index (HI), Average Wiener index (AWI), Modified wiener index (MWI), Schultz index (SI), and Gutman index (GI) with illustrations. Various examples are given for each of these topological indices and the necessary condition for the equality of connectivity index (CI) and WI is introduced.We have shown that if two graphs are isomorphic, then their WI are equal. Finally, we give an application of the WI, which is used as a simple screening model based on risk factors in patients to find the most effective factor in cancer.
Keywords: Vague set, vague graph, Wiener index, Zagreb index, medical screening, cancer
Mathematics Subject Classification: 05C99, 03E72.