Characterization of some alternating groups by order and largest element order

The prime graph (or Gruenberg-Kegel graph) of a finite group is a well-known graph. In this paper, first, we investigate the structure of the finite 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.