Chordality of Enhanced Power Graphs of Finite Simple Groups: A Resolution of a Conjecture

The enhanced power graph \PE(G) of a finite group $G$ has vertex set $G$, two elements being adjacent if and only if they generate a cyclic subgroup. We determine the finite simple groups $G$ for which \PE(G) is chordal: they are the cyclic groups of prime order, \Alt(7), \PSL_2(q) ($q\ge4$), \Sz(q) ($q=2^{2n+1}\ge8$) and \PSL_3(4). This answers an open problem of Bubboloni, Fumagalli and Praeger \cite{BFP}. We also prove that for every finite group $G$, if the power graph $P(G)$ is a cograph (resp.\ chordal) then so is \PE(G); the cograph case is a conjecture of Bubboloni, Fumagalli and Praeger \cite{BFP}.