Enhanced power graphs of finite groups with cograph structure

The enhanced power graph, $\mathcal{E}(G)$, of a group $G$ has vertex set $G$ and two elements are adjacent if they generate a cyclic subgroup. In the case of finite groups, we identify some striking and unexpected properties of these graphs, as well as links between properties of $\mathcal{E}(G)$ and properties of the group $G$. We prove that if $\mathcal{E}(G)$ is a cograph then it is also a chordal graph. Making use of properties of simplicial vertices, we characterise the finite groups $G$ whose enhanced power graph is diamond-free or a block graph. We also characterise the finite groups having enhanced power graph a cograph or a quasi-threshold graph, and those with $C_4$-free enhanced power graph. We use these characterisations to classify the finite nonabelian simple groups whose enhanced power graph is a cograph and give information on the finite simple groups whose enhanced power graph is $C_4$-free. Some open problems are posed.