Vertex Order Coloring in Fuzzy Graphs and its Applications

Data analytics and networking are inherently interdependent, as effective data collection, transmission, and processing rely on robust and optimally configured network infrastructures. Real-world networks are often characterized by uncertainty, vagueness, and dynamic constraints, necessitating advanced mathematical tools beyond classical graph-theoretic approaches. This book presents a comprehensive framework for optimal network analysis using vertex order coloring under fuzzy, intuitionistic fuzzy, and neutrosophic graph environments. Novel methodologies based on fuzzy vertex order coloring (FVOC), intuitionistic fuzzy vertex order coloring (IFVOC), and neutrosophic vertex order coloring are developed to identify and classify α-strong, β-strong, and γ-strong vertices, enabling systematic analysis and comparison of network robustness and efficiency. Various graph product operations, including co-normal, modular, residue, maximal, and related products are rigorously analyzed to determine optimal network configurations using key performance metrics such as chromatic number, distribution and weight of strong vertices, and minimum spanning tree weight.