Integrity on m -Polar Fuzzy Graphs and Its Application

Integrity for crisp graph theory is a well-defined topic. However, the integrity concept for fuzzy graphs has only recently been defined and extensively researched. However, in m -polar fuzzy graphs ( m PFG), each node as well as edges has m components. So, defining integrity in the m PF environment needs a new concept. As in the m -polar fuzzy environment, each node and edge has m components, so we have more flexibility to address the uncertainty rather than fuzzy as well as other uncertain environments. In this article, we developed a brand-new idea known as node integrity on m PFG and went in-depth on a few of their related properties. We have thoroughly covered some of their related properties as well as a brand-new idea called dominating integrity on m PFG. Different types of integrity on m PFG such as node integrity, dominating integrity, and edge integrity are discussed thoroughly along with some of its interesting facts have been introduced. Under isomorphism, their properties have also been studied. We also discussed the interrelation between them. A new type of m PFG called efficient m PFG which is directly related to dominating integrity concept has also been introduced. Several facts about efficient m PFG have also been studied here along with details descriptions. Finally, a real-world mobile network application that is directly related to the integrity of the m PFG concept has been discussed.