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 $rs \in ω(G)$ 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 $Γ(G)$ form a union of cliques. Moreover, we decide when a strongly regular graph is isomorphic to the Gruenberg-Kegel graph of a finite group. Besides this, we prove that a complete bipartite graph with each part of size at least $3$ can not be isomorphic to the Gruenberg-Kegel graph of a finite group.