On Recognition of the Sporadic Simple Groups $ HS $, J₃, $ Suz $, O^N,$ Ly $, $ Th $, Fi₂₃, and Fi₂₄^ by the Gruenberg–Kegel Graph

The Gruenberg–Kegel graph (the prime graph) of a finite group G is the graph whose vertices are the prime divisors of the order of G and two different vertices p and q are adjacent if and only if G contains an element of order $ pq $ . We find all finite groups with the same Gruenberg–Kegel graph as S for each of the sporadic groups S isomorphic to $ HS $ , J₃ , $ Suz $ , O^N , $ Ly $ , $ Th $ , Fi₂₃ , or Fi₂₄^ . In particular, we establish the recognition by the Gruenberg–Kegel graph for these eight groups S .