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 $\Gamma(G)$ 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.