Pythagorean fuzzy incidence graphs with application in illegal wildlife trade

Chemical engineers can model numerous interactions in a process using incidence graphs. They are used to methodically map out a whole network of interconnected processes and controllers to describe each component's impact on the others. It makes it easier to visualize potential process paths or a series of impacts. A Pythagorean fuzzy set is an effective tool to overcome ambiguity and vagueness. In this paper, we introduce the concept of Pythagorean fuzzy incidence graphs. We discuss the incidence path and characterize the strongest incidence path in Pythagorean fuzzy incidence graphs. Furthermore, we propose the idea of Pythagorean fuzzy incidence cycles and Pythagorean fuzzy incidence trees in Pythagorean fuzzy incidence graphs and give some essential results. We illustrate the notions of Pythagorean fuzzy incidence cut vertices, Pythagorean fuzzy incidence bridges, and Pythagorean fuzzy incidence cut pairs. We also establish some results about Pythagorean fuzzy incidence cut pairs. Moreover, we study the types of incidence pairs and determine some crucial results concerning strong incidence pairs in the Pythagorean fuzzy incidence graph. We also obtain the characterization of Pythagorean fuzzy incidence cut pairs using $ \alpha $-strong incidence pairs and find the relation between Pythagorean fuzzy incidence trees and $ \alpha $-strong incidence pairs. Finally, we provide the application of Pythagorean fuzzy incidence graphs in the illegal wildlife trade.