Finite groups with the given condition on the prime graph

The prime graph (or the Gruenberg–Kegel graph) of a finite group G is a simple graph GK(G) whose vertices are the prime divisors of the order of G, and two distinct vertices p and q are adjacent in GK(G) if and only if G contains an element of order pq.
The concept of Gruenberg–Kegel graph is very useful in finite group theory and in algebraic combinatorics.
In this talk, we discuss results on finite groups with the given condition on the prime graph (the Gruenberg-Kegel graph). In the “Kourovka Notebook”, A.V. Vasiliev posed question 16.26: Does there exist a natural number k such that no k pairwise nonisomorphic finite nonabelian simple groups can have the same prime graph? Conjecture: k = 5.
We discuss author’s results obtained on A.V. Vasiliev’s Conjecture. We also consider other results about the prime graph (the Gruenberg-Kegel graph).