Fuzzy graphs

In this thesis, fuzzy graphs are discussed and important results obtained on this subject have been compiled from various sources. A graph generally represents certain relationships between different elements in a set V. This gives an idea about the extent of the relationships between any two elements of the set V. If the graph is a weighted graph and all the weights between any two vertices are known, an inference can be made about the relationship between any two elements in the set V. However, in real life, it is not possible to know these weights precisely, and these weights are generally in fuzzy form. Therefore, it is more appropriate to use fuzzy relations for the relations between the elements of the set V. For example, if V represents certain regions and a network of roads is to be constructed between elements of V, then the costs of construction of the roads are not an exact number, but a fuzzy number. Hence, it will be much more realistic to use fuzzy graphs in real-life problems. Although Kaufman gave the first definition of fuzzy graph, it was Rosenfeld and Yeh and Bang who laid the foundations of fuzzy graph theory. In the study, firstly, after giving preliminary information about fuzzy sets, fuzzy relations, and classical graphs, the definition and basic concepts of fuzzy graphs are listed and the concept of connectedness in fuzzy graphs is examined. Then, fuzzy forests, fuzzy trees, and fuzzy cut sets are defined and their basic properties are studied. Finally, fuzzy bridges, fuzzy cutvertex, and fuzzy blocks are discussed and various results characterizing these concepts are given