(untitled) (European Journal of Pure and Applied Mathematics, 2025)
This research study delves into the spectral characteristics and energy measures associated with fuzzy graphs. We present the fuzzy geodetic spectrum and fuzzy detour spectrum, which capture the spectral properties of fuzzy graphs under geodetic and detour constraints. In addition, we propose and derive novel expressions for the fuzzy geodetic-Laplacian and detour-Laplacian energies, incorporating both upper and lower bounds. Towards the end, a real-world application of fuzzy geodetic-Laplacian energy pertaining to illegal immigration and human trafficking is described. It emphasizes its potential to improve decision-making measures, allowing more effective and coordinated crime prevention and resolution initiatives.