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

The enhanced power graph (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 (G) is chordal: they are the cyclic groups of prime order, (7), ₂(q) (q≥4), (q) (q=2²ⁿ⁺¹≥8) and ₃(4). This answers an open problem of Bubboloni, Fumagalli and Praeger {BFP}. We also prove that for every finite group G, if the power graph $P(G)$ is a cograph (resp.\ chordal) then so is (G); the cograph case is a conjecture of Bubboloni, Fumagalli and Praeger {BFP}.