The prime graph (or Gruenberg-Kegel graph) of a , nite group is a well-known graph. In this paper, , rst, we investigate the structure of the , nite groups with a non-complete prime graph. Then as an application, we prove that every alternating group An, where n ,31 is determined by its order and its largest element order. Also, we show that A32 is not characterizable by order and the largest element order.