The Divisibility Graph for F-Groups

A graph $D(G)$ is called the divisibility graph of $G$ if its vertex set is the set of noncentral conjugacy class sizes of $G$ and there is an edge between vertices $a$ and $b$ if and only if $a|b$ or $b|a$. We determine the number of connected components of the divisibility graph $D(G)$ when $G$ is an F-group. A finite group $G$ is called an F-group if for every $x, y \in G\setminus Z(G)$, $C_{G}(x)\leq C_{G}(y)$ implies $C_{G}(x)=C_{G}(y)$. We also prove that if the divisibility graph $D(G)$ in which $G$ is an F-group is a $k$-regular graph, then the divisibility graph $D(G)$ is a complete graph with $k+1$ vertices.