For a finitely generated group G the Nielsen graph Nn(G), n≥rank(G), describes the action of the group AutFn
Nearby in the stack
of automorphisms of the free group
Fn
on generating
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
AutFn
,
n≥4
. We prove nonamenability of Nielsen graphs
Nn(G)
for all
n≥max{2,rank(G)}
when
G
is indicable, and for
n
big enough when
G
is elementary amenable. We give an explicit description of
Nd(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.