On combinatorial properties of Gruenberg–Kegel graphs of finite groups

If G is a finite group, then the spectrum ω ( G ) is the set of all element orders of G . The prime spectrum π ( G ) is the set of all primes belonging to ω ( G ) . A simple graph Γ ( G ) whose vertex set is π ( G ) and in which two distinct vertices r and s are adjacent if and only if r s ∈ ω ( G ) is called the Gruenberg–Kegel graph or the prime graph of G . In this paper, we prove that if G is a group of even order, then the set of vertices which are non-adjacent to 2 in Γ ( G ) forms a union of cliques. Moreover, we decide when a strongly regular graph is isomorphic to the Gruenberg–Kegel graph of a finite group.