The prime graph of a sporadic simple group

Let G be a finite group and S a sporadic simple group. We denote by pi(G) the set of all primes dividing the order of G. The prime graph Gamma(G) of G is defined in the usual way connecting p and q in pi(G) when there is an element of order pq in G. The main purpose of this paper is to determine finite group G satisfying Gamma(G) = Gamma(S) (See Theorem 3) and to give applications which generalize Abe (Abe, S. Two ways to characterize 26 sporadic finite simple groups. Preprint) and Chen (Chen, G. (1996). A new characterization of sporadic simple groups. Algebra Colloq. 3:49-58). The results are elementary but quite useful.