Some results on the main supergraph of finite groups

Let G be a finite group. The main supergraph S(G) is a graph with vertex set G in which two vertices x and y are adjacent if and only if o(x) vertical bar o(y) or o(y) vertical bar o(x). In this paper, we will show that G congruent to PSL(2, p) or PGL(2, p) if and only if S(G) congruent to S(PSL(2, p)) or S ( PGL(2, p)), respectively. Also, we will show that if M is a sporadic simple group, then G congruent to M if only if S(G) congruent to S (M).