Bipolar Fuzzy Sets and Bipolar Fuzzy Graphs

In this chapter, we first review the notion of bipolar fuzzy sets and present several basic concepts concerning bipolar fuzzy graphs and bipolar fuzzy digraphs. We discuss different methods of construction of bipolar fuzzy graphs and their isomorphism properties. We describe certain types of bipolar fuzzy graphs, bipolar fuzzy walk, bipolar fuzzy bridge, strength of connectedness, weak and strong bipolar fuzzy edges. We establish the relations on bipolar fuzzy graphs, complement of bipolar fuzzy graphs, and crisp graphs with different operations, \(\alpha -\)cuts and \((\alpha ,\beta )-\)cuts. We also study certain operations and properties of complex bipolar fuzzy graphs. Moreover, with the help of composition of bipolar fuzzy relations, connectivity, and weighted matrices, we study the importance of bipolar fuzzy digraphs with a number of real-world problems. This chapter is basically adapted from [1, 2, 3, 45, 46, 51].