Advancements in Perfect Matchings within Neutrosophic Fuzzy Graphs: Theory and Applications

Graph theory has been widely used in exemplifying relational structure, and a greater generalization to fuzzy and neutrosophic space enables the exemplification of uncertainty, indeterminacy, and inconsistency in complex systems. Some of these extensions include the neutrosophic fuzzy graphs that provide a more detailed description of the loose relations between the vertices and the edges. However, unlike in classical graph theory, where the concepts of matching and perfect matching are well developed, very little has been studied on how the two concepts can be extended to neutrosophic fuzzy graphs. To seal this gap, the current paper develops and defines the so-called perfect matching in neutrosophic fuzzy graphs. We provide our strict definitions, structural properties, and neutrosophic fuzzy matching numbers of different graph structures. It has been compared with the classical matching theory to demonstrate the influence of the presence of indeterminacy on matching behavior on an uncertain basis. The findings draw on theoretical knowledge and practical implications, particularly where it concerns the problem of uncertain assignments and network allocation in real life, such as in the case of decision-making systems, optimization, and intelligent network analysis.