On transitivity and (non)amenability of Aut(F_n) actions on group presentations

For a finitely generated group $G$ the Nielsen graph $N_n(G)$, $n\geq \operatorname{rank}(G)$,describestheactionofthegroup$\operatorname{Aut}F_n$ofautomorphismsofthefreegroup$F_n$ongenerating$n$-tuples of G by elementary Nielsen moves. The question of (non)amenability of Nielsen graphs is of particular interest in relation with the open question about Property $(T)$ for $\mathrm{Aut}{F}_n$, $n\ge 4$. We prove nonamenability of Nielsen graphs $N_n(G)$ for all $n\ge \max{2,\operatorname{rank}(G)}$when$G$isindicable,andfor$n$ big enough when $G$ is elementary amenable. We give an explicit description of $N_d(G)$ for relatively free (in some variety) groups of rank $d$ and discuss their connectedness and nonamenability. Examples considered include free polynilpotent groups and free Burnside groups.