On Some Types of Pythagorean Fuzzy Labelings

Abstract

With the extensive application of networks in every field of science such as engineering, chemistry, biology, and computer networking, the modeling of complex uncertain systems has become more prominent. Graph theory is one of the efficient mathematical methods of representing such systems in which issues of real life are described with the assistance of vertices and edges. Classical graphs are not satisfactory in models involving vagueness and indecision. Fuzzy graphs and their extensions have been developed to address these issues for their handling. Intuitionistic fuzzy graphs (IFGs) on the basis of both membership and non-membership grades provided a more versatile structure to handle uncertainty. However, the condition that the sum of membership and non-membership cannot be more than one puts a limit to their expressiveness. To solve this, Pythagorean fuzzy graphs (PFGs) have been introduced so that the sum of squares of such degrees can be less than or equal to one, thus giving a broader and more precise representation of uncertainty. In This thesis we solves the issue of converting IFGs to PFGsbydevelopingamathematicalframeworkthat preserves the graph structure but enhances its fuzzy representation capability. By using super edge magic labeling we assign unique la bels to the vertices and edges and sum should be constant. The transformation from IFG to PFGis carried out systematically. Additionally, the research explores the consequences of this change in super edge magic labeling, introducing new information on the labeling structures that can be employed on PFGs. The findings prove that PFGs not only generalize IFGs but also are more flexible and effective in graph-based modeling, particularly in scenarios where greater uncertainty and intricate relationships exist.