Some Topological Indices in Fuzzy Graphs

Abstract

Fuzzy graph is one of the application tool in the field of mathematics, which allow the users easily describe the relation between any notions because the nature of fuzziness is favorable for any environment. Fuzzy graphs are beneficial to give more precision and flexibility to the system as compared to the classical models. A topological index is a numerical quantity for the structural graph of a molecule. Generally, the topological indices are familiar in chemistry but a graph structure is from the mathematical background and Harold Wiener, who developed it in his theory and routed to construct a huge branch of chemical graph theory. Usage of Wiener indices is to find the properties of the type of alkanes known as paraffin. Moreover, its application is not only in the field of chemistry and applied in all areas including computer science, networking etc. Lot of topological indices are available in graph theory and its transliterated to field of fuzzy graphs. In crisp sense, the vertices and edges are having the membership value as 1 but in the area of fuzzy graphs both vertices and edges are important with distinct memberships. This make complete difference from the crisp graph. The topological indices exist in crisp but new to the fuzzy graph environment. This paper explains about the indices in fuzzy graph theory. In this paper, we discuss about new term definitions on Zagreb indices, Harmonic index, Randic index, Estrada index and Padmakar–Ivan index. Bounds for Zagreb indices are proved. Algorithms for each index are shown. Finally, two applications on human trafficking and internet routing are discussed.