A novel approach on laplacian energy of σJ-dominating complex spherical fuzzy graph structures and with application

The CSFS is an extension of CPFS and CPicFS. The design addresses vagueness more explicitly. It extends the real subset’s range to complex with the use of unit disc. Due to its coverage in three directions: complex membership functions, complex abstinence membership function, and complex non-membership functions, CSFS has more flexibility than CFSs, CIFSs, CPFSs, and CPicFS. It can be applied in various ways through fuzzy control. The primary objective of this study is to present CSFGS, which exhibits exceptional performance in tackling challenging problem, particularly ones that involve multiple relationships. GSs and CSFSs are combined in this concept. As a result, many problems are combined with different relations. The AM of CSFGS and degree of vertex presence have been found. The lower and upper constraints on the energy of CSFGS are derived by calculating the energy of CSFGS. Furthermore, we have provided some details regarding the LE of the σ J -dominant CSFGS. We also propose the notion of isomorphic and identical σ J -dominating CSFGS. Finally, we have discussed a real-world example based on temperature variation and climatic data analysis in an environment with a LE of the σ J -dominant CSFGS, to demonstrate the applicability of the generated results. Also, an algorithm has been developed to clarify the fundamental workings of our application.