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.

Nonlinear metamaterials have interesting properties that could change the way technology works. Modelling the complex behaviour of metamaterials is needed to improve their performance and make new gadgets. This chapter looks at fuzzy logic models for complex effects in metamaterials. This part talks about fuzzy sets, membership functions, fuzzy rules, and inference systems. Metamaterials don't behave in a straight way, so there are problems with describing them. Input factors, membership functions, along with fuzzy rules are talked about in the modelling method of fuzzy logic for nonlinear metamaterials. Acoustic and visual metamaterial case studies show how fuzzy logic models can be used. This part also talks about the latest developments, possible uses, and pros and cons of fuzzy logic models in nonlinear metamaterials. This chapter tells scientists and engineers how to use fuzzy logic to model and understand nonlinear metamaterials.

Fuzzy Linear Programming (FLP) has the potential to be used in optimizing economic planning and making decisions under uncertainty. FLP incorporates fuzzy logic into linear programming to represent and manage economic parameters that are uncertain, e.g., costs, profits, and availability of inputs. Practical applications in Economic resource allocation illustrate the effectiveness of FLP, as demonstrated by the study. ISMC-based FLP has been shown to offer a flexible solution that is more adaptable and realistic than classical linear programming models. This research reiterates practical economic scenarios through fuzzy data, considering uncertainties and vagueness in risk measures, helping to make better and effective decision-making. Future research directions involve combining FLP with Artificial Intelligence (AI) and Big Data in finance to improve its utility in complex and dynamically changing economic systems, allowing easier and automatic decision making. FLP moves beyond the deterministic nature of traditional modelling by integrating fuzzy data, allowing the model to reach more flexible and realistic results and providing better decision-making.