Topological Indices in Fuzzy Graphs -An Overview

Abstract

Graph theory provides a mathematical framework for representing systems with binary relations.  This theory is crucial for modeling and analyzing various systems where relationships between elements can be depicted as connections or edges between nodes or vertices paving way for numerous applications in designing communication networks, VLSI circuits, and planning transport networks. Fuzzy set is a powerful tool to capture vagueness and uncertainty through membership functions. Incorporation of fuzzy sets to graph theory forms the basis for fuzzy graphs, where the uncertainty of the vertices and edges are captured through membership values.  In chemical science, a topological representation of the molecule is called the molecular graph.  A topological index is a numerical quantity for the structural graph of the molecule of a chemical compound, with collection of atoms representing the atoms in the molecule and the chemical bonds representing the chemical bonds considered as edges.  This numerical measure finds applications beyond chemistry and has evolved as one of the powerful tools for decision making.  Many topological indices existing in crisp graphs are extended to fuzzy graphs with proven applications.  Some topological indices in fuzzy graphs are Wiener index, Modified Wiener index, Hyper Wiener index, Schultz index, Gutman index, Zagreb index, Hyper Zagreb index, Harmonic index and Radic index.  A comprehensive review of the topological indices in fuzzy graphs with potential applications is presented in this work with a discussion on the scope for future work.