Product of Total and Range Fuzzy Edge Labeling of Interval-Valued Fuzzy Graph

The fuzzy graph theory can be described as a generalization of the classical graph theory wherein the structures of the graphs are defined in uncertain form. Interval-valued fuzzy graphs are generalizations of fuzzy graphs wherein the membership value of each structure is given in terms of intervals instead of crisp values. However, the existing approaches for labeling edges in fuzzy graphs are mainly either additive or rule-based, and thus they cannot efficiently accommodate complex networks with high uncertainties. To address the challenge, the authors present a new method known as Product of Total and Range Fuzzy Edge Labeling (PTARFEL) for interval-valued fuzzy graphs. The PTARFEL approach applies multiplicative procedure to calculate the membership values of the edges according to the total and range of the membership values of the connected vertices. Interval valued ε neighborhood construction is employed to provide appropriate labeling and to keep the intervals of vertices and edges distinct under various graph settings. The theoretical properties are developed using rigorous proof concerning the fuzzy graph, the interval-valued labeled graph, and the conditions that guarantee consistency of the product-based labeling. The effectiveness of the proposed labeling approach is evaluated for the well-known graph classes including path, star, and cycle graphs. The simulation results confirm the ability of the proposed label to satisfy the fuzzy conditions, different label allocation, and consistency even with perturbation of intervals; hence, improving the performance of the conventional labeling approach. In conclusion, the proposed model is a generalized and flexible framework to represent uncertainty in network structures. Therefore, it generalizes the concept of interval-valued fuzzy graph labeling. This model enables its applicability in decision making, communication networks, and optimization problems where the information is always imprecise.