On the multiplicity order of spinnable star-like transformation semigroup Tw*n

The application of graph theory has gained significant traction within the realm of the algebraic theory of semigroups. This study delves into exploring the geometric properties of the star-like transformation semigroup \alpha\omega_n^*, a distinctive category of transformation, and delineates a tropical graph (curve) by elucidating its algebraic and tropical structure. Through this investigation, various tropical properties are established, offering insights into the graph theory aspects of star-like spinnable T\omega_n^* transformation semigroups. Consequently, the objective of this research is to delineate and characterize several tropical and combinatorial functions applicable to T\omega_n^*.