Let G be a finite group and πe(G) be the set of element orders of G. Let k ∈ πe(G) and m k be the number of elements of order k in G. Set

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)∣o(y) or o(y)∣o(x). In this paper, we will show that G≅PSL(2,p) or PGL(2,p) if and only if S(G)≅S(PSL(2,p)) or S(PGL(2,p)), respectively. Also, we will show that if M is a sporadic simple group, then G≅M if only if S(G)≅S(M).