A Novel Framework for Eigenvalue-Based Vertex Importance and Stability Analysis

Spectral graph theory has emerged as a powerful tool for analyzing structural properties of complex networks, while fuzzy graph theory provides a flexible framework to model uncertainty and imprecision inherent in real-world systems. In this paper, we develop a unified theoretical framework for spectral fuzzy graphs by integrating fuzzy membership concepts with spectral matrix analysis. We introduce a novel fuzzy spectral importance function that quantifies the significance of vertices by combining eigenvalue contributions with vertex membership strengths. Fundamental properties of this measure are established through rigorous propositions, including invariance under null membership and uniformity in symmetric fuzzy structures. Furthermore, we derive new theoretical results on spectral dominance and perturbation stability, providing bounds that characterise the behaviour of fuzzy spectral measures under structural variations. The proposed framework extends classical spectral concepts such as adjacency spectra, Laplacian eigenvalues, and algebraic connectivity to fuzzy environments, thereby enabling deeper insights into uncertain network structures. The results presented in this study not only advance the mathematical foundations of spectral fuzzy graph theory but also open avenues for applications in complex network analysis, decision-making systems, and data-driven modelling under uncertainty.