Certain operations on interval-valued picture fuzzy graphs with application

Graph theory has various applications in computer science, such as image segmentation, clustering, data mining, image capturing, and networking. Fuzzy graph (FG) theory has been widely adopted to handle uncertainty in graph-related problems. Interval-valued picture fuzzy graphs (IVPFGs) are a generalization of FGs, interval-valued FGs, intuitionistic fuzzy graphs (IFGs), and interval-valued IFGs. This paper introduces the concept of interval-valued picture fuzzy sets to graph theory and presents a new type of graph called the IVPFG. Within this framework, we define the degree, order, and size of IVPFGs. The paper further explores various operations on IVPFGs, including the Cartesian product, composition, join, and union. The paper delves into the properties of these operations, providing proofs and examples to support the findings. By studying the operations on IVPFGs, we can gain insights into their behavior and leverage this knowledge for solving graph-based problems in the presence of uncertainty. Also, an application regarding merging of community is provided.