Abstract
Our study presents a mathematical framework for modelling and analysing intuitionistic fuzzy graphs through matrix representations and spectral analysis. Extending fuzzy-set theory, we integrate membership and non-membership degrees to capture uncertainty and hesitation. We introduce intuitionistic fuzzy adjacency, incidence, and Laplacian matrices, derive spectral bounds that generalize the Perron-Frobenius theorem, and prove these bounds using variational principles, matrix norm inequalities, and perturbation techniques, demonstrating that eigenvalues are bounded by aggregated degrees. We validate our theoretical findings with computational experiments and case studies on simulated social and organizational networks, using Python to visualize the algebraic connectivity of the Laplacian as a resilience metric. We discuss practical implications for network robustness and resilience analysis. By modelling dual aspects such as trust and distrust, our approach deepens insights into decision-making systems, control mechanisms, and biological networks. These contributions lay the groundwork for dynamic and higher-order intuitionistic fuzzy-graph research across diverse application domains.