A Spherical Fuzzy Planar Graph Approach to Optimize Wire Configuration in Transformers

Hao Guan ¹ Saira Hameed ^2* Sadaf . ² Aysha Khan ³ Jana Shafi ³

• ¹ Guangzhou University, Guangzhou, Guangdong Province, China
• ² University of the Punjab, Lahore, Pakistan
• ³ Prince Sattam Bin Abdulaziz University, Al-Kharj, Saudi Arabia

In this modern era, graph theory has become an integral part of science and technology. It has enormous applications in handling various design-based problems. In this study, we present a new approach that increases the ability of graph theory to deal with uncertain challenges. Spherical fuzzy graph addresses the uncertainty domain more broadly, going beyond classical fuzzy graphs other generalized fuzzy graph types. The concept of spherical fuzzy planarity value provides a new method to evaluate the edge intersection and order in the graph. The ability of spherical fuzzy planar graph (SF PG) to model complex relationships can enable more precise and reliable network designs and analyses. In this work, the concepts of spherical fuzzy multigraph (SF MG), spherical strong and weak edges, and the value of planarity for spherical fuzzy graphs (SF Gs) are introduced. Moreover, the concept of the degree of planarity within the context of spherical fuzzy planar graphs, and the notion of strong and weak faces are introduced. Additionally, we delve into the construction of spherical fuzzy dual graphs, which can be realized in cases where the fuzzy graph is planar or possesses a degree of planarity ≥ 0.67. This notion also serves as the foundation for certain basic theorems. We talk about some significant findings