On Recognition of the Sporadic Simple Groups , , , , , , , and by the Gruenberg–Kegel Graph

The Gruenberg–Kegel graph (the prime graph) of a finite group is the graph whose vertices are the prime divisors of the order of and two different vertices and are adjacent if and only if contains an element of order . We find all finite groups with the same Gruenberg–Kegel graph as for each of the sporadic groups isomorphic to , , , , , , , or . In particular, we establish the recognition by the Gruenberg–Kegel graph for these eight groups .