Abstract
Today, fuzzy graphs (FGs) have a variety of applications in other fields of study, including medicine, engineering, and psychology, and for this reason, many researchers around the world are trying to identify their properties and use them in computer sciences as well as finding the smallest problem in a network. The concept of a Cayley fuzzy graph has become a standard part of the toolkit used to investigate and describe groups. Also, Cayley fuzzy graphs are good models for interconnection networks, and they are useful in semigroup theory for establishing which elements are l and R related. The previous definition limitations in the FGs have directed us to offer a new classification in terms of Cayley fuzzy graphs. So, in this paper, two new definitions of Cayley fuzzy graphs (CFGs) and pseudo-Cayley fuzzy graphs (PCFGs) are discussed and their rough approximations are studied. Also, some properties of fuzzy rough sets (FRSs) in CFGs and PCFGs have been investigated. Finally, we presented the determination of the most effective person in the Water and Sewerage Organization and the importance of using refereeing facilities in football matches between club teams, by using CFG in the presented applications.