Journal of Advances in Mathematics and Computer Science

This paper develops a unified mathematical framework to analyze and compare the structural properties of Pythagorean fuzzy graphs (PFGs) under threshold conditions, using vertex degree-based methods. A dual thresholding mechanism is introduced, based on membership and non-membership degrees, to construct reduced subgraphs that preserve structurally significant vertices and edges. In the first part, the concept of reduced degree in pythagorean fuzzy graphs is defined and analyzed, with a focus on how vertex degrees change under threshold filtering. Several theorems are established to characterize the behavior of reduced graphs, particularly in relation to monotonicity, boundedness, and structural simplification.

In the second part, degree-based similarity measures-including cosine, Euclidean, Gaussian, and Manhattan similarities are introduced to evaluate the closeness between filtered versions of PFGs. These similarity indices are derived from degree vectors and are shown to satisfy key properties such as identity, symmetry, and boundedness. Comparative analysis, supported by examples and tabular evaluations, illustrates the effect of threshold variation on similarity scores. The proposed dual-concept approach enhances the understanding of PFG structure under filtering and offers a robust mathematical tool for structural comparison in fuzzy systems, with potential applications in image segmentation, decision support, and network reduction.