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The prime graph of a finite group $G$ is the labelled graph $\Gamma(G)$ withvertices the prime divisors of $|G|$ and edges the pairs $\{p,q\}$ for which$G$ contains an element of order $pq$. A group $G$ is recognisable by its primegraph if every group $H$ with $\Gamma(H)=\Gamma(G)$ is isomorphic to $G$.Cameron and Maslova have shown that every group that is recognisable by itsprime graph is almost simple, which justifies the significant amount ofattention that has been given to determining which simple (or almost simple)groups are recognisable by their prime graphs. This problem has been completelysolved for certain families of simple groups, including the sporadic groups. Anatural extension of the problem is to determine which groups are recognisableby their unlabelled prime graphs, i.e. by the isomorphism types of their primegraphs. There seem to be only very limited results in this direction in theliterature. Here we determine which of the sporadic finite simple groups arerecognisable by the isomorphism types of their prime graphs. We also show thatfor every sporadic group $G$ that is not recognisable by the isomorphism typeof $\Gamma(G)$, there are infinitely many groups $H$ with $\Gamma(H) \cong\Gamma(G)$.

Recognisability of the sporadic groups by the isomorphism types of their prime graphs