The aim of this work is the understanding of the impact of acido-basic properties of oxide supports on the metal-support interaction as well as on the effects of reactive conditions on the structural, electronic, thermodynamic and reactive properties of hyper-dispersed metallic catalysts. The chosen model catalyst is a monometallic platinum cluster supported on alumina which is the reference of fundamental studies to improve the understanding of this complex system. DFT calculations were performed from periodic models to determine the surface state of the catalyst under representative experimental temperature and pressure. The influence of nano-clusters' size combined with the presence of chlorine on the thermodynamic stability of supported Pt clusters on γ-alumina demonstrated a significant stabilization of Pt3 clusters with the migration of surface species (H, OH and Cl) on the particle. The morphology reconstruction from biplanar (the most stable in absence of hydrogen) to a cuboctahedral morphology for the high hydrogen coverages on the supported Pt13 cluster is demonstrated and explains several experimental observations still controversial. The impact of the reactive conditions on the C-C and C-H bond scission from ethane on the Pt13 cluster supported on γ-alumina was demonstrated. Results helps to better understand the importance of temperature and pressure conditions in the reactive medium (notably the P(H2)/P(C2H6) ratio) on the stability of reactive intermediates. These results are in agreement with experimental observations and highlight that the optimum of hydrogen pressure is relatively high in the case of catalytic reforming reactions catalyzed by Pt only.