Novel Spectral Conditions for Diagonalizability and Connectivity in Spectral Fuzzy Graph Theory

This paper explores the properties of fuzzy matrices in fuzzy graphs and the conditions for the diagonalizability of fuzzy matrices. Necessary and sufficient conditions for fuzzy graphs to have non-negative and distinct eigenvalues are provided, and the existence of orthogonal eigenvectors corresponding to distinct eigenvalues in fuzzy matrices are discussed. Also, conditions for the second smallest eigenvalue of the Laplacian matrix are established to ensure connectivity in fuzzy graphs.