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(Abridged). The gamma-ray emission from galaxy clusters hosting activegalaxies is a complex combination of diffuse and point-like emission withdifferent spectral and spatial properties. We discuss the case of the Perseuscluster containing the radio-galaxy NGC 1275 that has been detected as a brightgamma-ray source by the Fermi-LAT experiment. We provide a detailed study ofthe gamma-ray emission coming from the core of Perseus by modeling the centralAGN emission with a multiple plasma blob model, and the emission from thecluster atmosphere with both a Warming Ray (WR) model and Dark Matter (DM)neutralino annihilation models. We set constraints on both the central galaxyand cluster SED models by using multi-frequency data including the observationsobtained by Fermi and MAGIC. We find that: i) in all the viable models for thecluster gamma-ray emission, the emission detected by Fermi from the Perseuscore is dominated by the active galaxy NGC 1275, that is found in ahigh-emission state; ii) the diffuse gamma-ray emission of the cluster, in theWR model and in the DM models with the highest allowed normalization, could bedetected by Fermi if the central emission from NGC1275 is in a low-emissionstate; iii) Fermi can have the possibility to resolve and detect the diffusegamma-ray flux coming from the outer corona of the Perseus atmosphere at r> 800kpc. Our results show that a simultaneous study of the various emissionmechanisms that produce diffuse gamma-rays from galaxy clusters and thoseproducing gamma-rays from active galaxies residing in the cluster atmospheresis crucial first to disentangle the spectral and spatial characteristics of thegamma-ray emission and secondly to assess the optimal observational strategy inthe attempt to reveal the still elusive diffuse gamma-ray emission widelypredicted for the atmospheres of large-scale structures.

Disentangling the gamma-ray emission of NGC1275 and that of the Perseus cluster