Given a finite group [Formula: see text], the Engel graph of [Formula: see text] is a directed graph [Formula: see text] encoding pairs of elements satisfying some Engel word. Namely, [Formula: see text] is the directed graph, where the vertices are the non-hypercentral elements of [Formula: see text] and where there is an arc from [Formula: see text] to [Formula: see text] if and only if [Formula: see text] for some [Formula: see text]. From previous work, it is known that, except for a few exceptions, [Formula: see text] is strongly connected. In this paper, we give an absolute upper bound on the diameter of [Formula: see text], when [Formula: see text] is strongly connected.
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Lucchini et al. (2024) studied this question.
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