Criterion of unrecognizability of a finite group by its Gruenberg–Kegel graph

The Gruenberg-Kegel graph $\Gamma(G)$ associated with a finite group $G$ has as vertices the prime divisors of $|G|$, an edge from $p$ to $q$ if and only contains element order $pq$. This been subject much recent interest; one our goals here is give survey some this material, relating groups same graph. However, main aim prove several new results. Among them are following. - There infinitely many there $H$ non-trivial solvable radical such that $\Gamma(G)=\Gamma(H)$. function $F$ on natural numbers property $n$-vertex whose labelled by pairwise distinct primes more than $F(n)$ groups, then it groups. (The we satisfies $F(n)=O(n^7)$, but probably not best possible.) If $\Gamma$ finitely all almost simple; moreover, at least three non-adjacent vertices, $2$ odd vertex. Groups power graphs, or commuting isomorphic have ${^2}G_2(27)$ $E_8(2)$ uniquely determined isomorphism types their graphs. In addition, consider no edges. These in which every order, studied under name \emph{EPPO groups}; completing line research, complete list