21.23 (2026)

A graph is called a cograph if it has no induced subgraph isomorphic to a path with 4 vertices. A graph is said to be chordal if it has no induced cycles with $n$ vertices for every $n \geqslant 4$. For a finite group $G$, the enhanced power graph $\mathcal{E}(G)$ is the graph with vertex set $G$ and edges $\{x, y\}$ for all $x \neq y \in G$ such that $\langle x, y \rangle$ is cyclic.
$\qquad$ (a) For a given integer $n \geqslant 4$, determine the set of all finite nonabelian simple groups $G$ such that $\mathcal{E}(G)$ has no induced cycles with $n$ vertices.
$\qquad$ (b) Determine the set of all finite nonabelian simple groups $G$ such that $\mathcal{E}(G)$ is chordal.

In (Preprint, 2025, https://arxiv.org/abs/2510.18073) we proved that if the enhanced power graph of a given finite group is a cograph, then it is also chordal. Also the finite nonabelian simple groups whose enhanced power graph is a cograph are described, and additional information is obtained on finite nonabelian simple groups whose enhanced power graph has no induced cycles with 4 vertices.