On combinatorial properties of Gruenberg-Kegel graphs of finite groups

If G is a finite group, then the spectrum omega(G) is the set of all element orders of G. The prime spectrum pi(G is the set of all primes belonging to omega(G). A simple graph Gamma(G) whose vertex set is pi(G) and in which two distinct vertices r and s are adjacent if and only if rs is an element of omega(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 Gamma(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.