  We show that nonlocal minimal cones which are non-singular subgraphs outside
the origin are necessarily halfspaces.
  The proof is based on classical ideas of~\cite{DG1} and on the computation of
the linearized nonlocal mean curvature operator, which is proved to satisfy a
suitable maximum principle.
  With this, we obtain new, and somehow simpler, proofs of the Bernstein-type
results for nonlocal minimal surfaces which have been recently established
in~\cite{FV}. In addition, we establish a new nonlocal Bernstein-Moser-type
result which classifies Lipschitz nonlocal minimal subgraphs outside a ball.
