  We prove a few results about non-nilpotent graphs of symmetric groups $S_n$
-- namely that they have a Hamiltonian cycle and they satisfy a conjecture of
Nongsiang and Saikia. The latter is likewise proven for alternating groups
$A_n$. We also show that the class of non-nilpotent graphs does not have any
''local'' properties, ie. for every simple graph $X$ there is a group $G$, such
that its non-nilpotent graph has $X$ as an induced subgraph.
