The role of topological photon spheres in constraining the parameters of black holes

In this paper, we investigate the topological photon sphere from two distinct perspectives. In the first view, we examine the existence and characteristics of topological photon (anti -photon) spheres for black holes with different structures, such as Einstein-Young-Mills non -minimal, AdS black holes surrounded by Chaplyginlike dark fluid, and Bardeen -like black holes in Einstein-Gauss-Bonnet gravity. Furthermore, we delve into the deeper perspective of the necessity of photon spheres for super -compact gravitational structures such as black holes. By leveraging this necessity, we propose a classification of the parameter space of black hole models based on the existence and positioning of photon spheres. This approach enables the determination of parameter ranges that delineate whether a solution represents a black hole or a naked singularity. In essence, the paper illustrates the utility of the photon sphere as a notable test for establishing the permissible and non -permissible parameter ranges within specific theories of black hole solutions.