On combinatorial properties of Gruenberg–Kegel graphs of finite groups

Abstract

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 \(rs \in \omega (G)\) 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.