On the finite group whose proper enhanced power graph is claw-free

Abstract

Let $G$ be a finite group. The \emph{enhanced power graph} of $G$, denoted by $\mathcal{E}(G)$, is the graph with vertex set $G$ in which two vertices $u$ and $v$ are adjacent if and only if there exists an element $w \in G$ such that both $u$ and $v$ belong to $\langle w \rangle$. The \emph{proper enhanced power graph} of $G$, denoted by $\mathcal{E}^{**}(G)$, is the subgraph of $\mathcal{E}(G)$ induced by the non-dominating vertices. The main objective of this paper is to investigate finite groups whose proper enhanced power graph is claw-free, that is, contains no induced subgraph isomorphic to the complete bipartite graph $K_{1,3}$. We first prove that $\mathcal{E}(G)$ is claw-free if and only if $G$ is cyclic. The set of dominating vertices of $\mathcal{E}(G)$ forms a cyclic subgroup of the center of $G$, namely the \emph{cyclicizer} $\cyc(G)$ of $G$. This allows us to give a precise description of the structure of $G/\cyc(G)$ when $\mathcal{E}^{**}(G)$ is claw-free. If $G$ is solvable but not nilpotent, then $G$ is metacyclic, or $G/\cyc(G)$ is either a Frobenius group or a $2$-Frobenius group. If $G$ is non-solvable, then $G/\cyc(G)$ is isomorphic to $\PSL(2,q)$ or $\PGL(2,q),$ and this allows us to give a complete classification of the non-solvable groups whose proper enhanced power graph is claw-free.