Energy of Fuzzy graphs

Pythagorean fuzzy graphs (PFGs) extend traditional fuzzy graphs by incorporating both membership and non-membership values, offering greater flexibility in modeling uncertainty. This makes PFGs valuable in complex applications like decision-making, network analysis, and optimization. Graph energy, originally introduced in classical graph theory, is extended in PFGs by considering both membership and non-membership adjacency matrices, providing a comprehensive tool for analyzing uncertain systems.This review explores the mathematical formulation of PFG energy, including Laplacian energy, and discusses computational challenges such as eigenvalue computation. We highlight scalable methods for calculating PFG energy and compare PFGs with other fuzzy graph models, emphasizing their strengths in handling uncertainty. Additionally, we discuss future research directions, including extensions to Pythagorean fuzzy digraphs and hesitant Pythagorean fuzzy graphs. PFGs are shown to be increasingly relevant in fields like communications, AI, and environmental management