FUZZIFICATION OF CRISP GRAPHS AND ITS IMPLICATIONS

Traditional fuzzy graph models assign membership values to vertices and edges based on a specific uncertain situation. However, this work investigates a novel approach: representing the situation as a graph and deriving fuzziness from the graph's inherent structure. We introduce "ratio labeling" (RL), a new labelling procedure where vertex and edge membership grades are determined by graph parameters. These labels, derived directly from the graph's structure, characterize the graph itself and serve as the basis for examining the admissibility of fuzziness within the graph. This approach allows the study of fuzziness arising from the properties of the graph representing a situation. This paper explores this new idea and examines certain graphs for the admissibility of fuzziness. This topic study the methodologies, properties, and applications of ratio labelling in fuzzy graph identification, focusing on its theoretical foundations and practical implications in solving real-world problems. Furthermore, the proposed ideas are illustrated with several numerical instances. To emphasize the theoretical concept, an application that ensures an effective communication between groups of people in a social media under RL is discussed.