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