A spherical fuzzy planar graph approach to optimize wire configuration in transformers

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. The spherical fuzzy graph addresses the uncertainty domain more broadly, going beyond classical fuzzy graphs other than generalized fuzzy graph types. The concept of the spherical fuzzy planarity value provides a new method to evaluate the edge intersection and order in the graph. The ability of the spherical fuzzy planar graph <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="m1"><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mi mathvariant="script">S</mml:mi><mml:mi mathvariant="script">F</mml:mi><mml:mi mathvariant="script">P</mml:mi><mml:mi mathvariant="script">G</mml:mi></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:math> to model complex relationships can enable more precise and reliable network designs and analyses. In this work, the concepts of spherical fuzzy multigraph <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="m2"><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mi mathvariant="script">S</mml:mi><mml:mi mathvariant="script">F</mml:mi><mml:mi mathvariant="script">M</mml:mi><mml:mi mathvariant="script">G</mml:mi></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:math> , spherical strong and weak edges, and the value of planarity for spherical fuzzy graphs <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="m3"><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mi mathvariant="script">S</mml:mi><mml:mi mathvariant="script">F</mml:mi><mml:mi mathvariant="script">G</mml:mi><mml:mi>s</mml:mi></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:math> 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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="m4"><mml:mrow><mml:mo>≥</mml:mo><mml:mn>0.67</mml:mn></mml:mrow></mml:math> . This notion also serves as the foundation for certain basic theorems. We talk about some significant findings related to this subject. We discuss some major results linked to this topic. To show the worth and importance of our work, we also provide a real-world application.