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

Abstract

The \emph{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, $\Alt(7)$, $\PSL_2(q)$ ($q\ge 4$), $\Sz(q)$ ($q=2^{2n+1}\ge 8$) 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}.