Improving the Efficiency of Fuzzy Graphs and Their Complements Using Some Influencing Parameters

This study focuses on constructing optimal network structures for fuzzy graph (FG) products. In graph theory, the complement of a FG product is essential since it analyses alternate interactions between the vertices. Such a complement is used to represent situations in which specific connections are deliberately excluded, which helps to understand complex networks and recognise how situational changes impact the behaviour of the overall network. The study of FG products and their complement is the primary focus of this research, with particular emphasis on parameters such as domination number (DN), minimal spanning tree (MST) weight, and burning number Bn. This study examines how key parameters interact with graph products, such as modular and residue graph products, offering a comparison between the FG products and their complement. Additionally, a computational approach is introduced to visualise the concept using Python code. The DN determines vertex control and influence, whereas the MST weight is used to solve network optimisation problems. Furthermore, the study investigates the Bn in FG products, which spread information within the network. Python code for the graph product and its complement is used to validate the output for the given parameters. This study contributes to the area by providing a systematic way to understand and enhance graph structures using FG products and their complements.