On combinatorial properties of Gruenberg--Kegel graphs of finite groups
Mingzhu Chen, Ilya B. Gorshkov, Natalia V. Maslova, Nanying Yang
TL;DR
The paper investigates Gruenberg--Kegel graphs $Γ(G)$ of finite groups, examining how the element-order spectrum $ω(G)$ determines adjacency among primes in $π(G)$ with the goal of characterizing groups by their prime graphs. For even-order groups with $t(2,G)≥2$, it proves that the set of primes non-adjacent to $2$ in $Γ(G)$, denoted $τ$, is a union of cliques and provides a normal-series description with a solvable radical $K$ and a non-abelian simple socle $S$, yielding the bound $t(2,G)≤ t(2,S)$ (or $t(2,G)=2$ in certain cases). It further classifies strongly regular graphs that can arise as GK graphs, showing they must be either the complement to a triangle-free SRG or a complete multipartite graph with all parts of size $2$, and it rules out complete multipartite GK graphs with parts of size at least $3$. These results connect combinatorial GK-graph properties to structural group theory, aiding in the characterization of groups by their prime graphs and clarifying the boundary between solvable and non-solvable cases in the SRG GK-context.
Abstract
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.