Energy and Laplacian Energy of Pythagorean Intuitionistic Fuzzy Graphs with Applications in Medical Diagnosis Networks

This study extends fuzzy graph energy analysis by introducing energy and Laplacian energy for Pythagorean Intuitionistic Fuzzy Graphs (PIFGs), a powerful generalization of intuitionistic fuzzy graphs capable of representing higher degrees of uncertainty. A novel connection matrix for PIFGs is defined, and new formulations for energy and Laplacian energy are established, along with sharp lower and upper bounds. Beyond theoretical contributions, the approach is applied to medical diagnosis networks, where vertices represent symptoms, diagnostic tests, and diseases, and edges encode Pythagorean intuitionistic fuzzy relationships. These measures quantify both the overall strength of associations (energy) and their structural irregularity (Laplacian energy), offering interpretable indicators for diagnostic certainty or ambiguity. The framework provides a robust mathematical basis for decision-making in biomedical contexts where data are uncertain, imprecise, or conflicting.