  Generalizing the star graphs of M\"uller-Stach and Westrich, we describe a
class of graphs whose associated graph hypersurface is equipped with a
non-trivial torus action. For such graphs, we show that the Euler
characteristic of the corresponding projective graph hypersurface complement is
zero. In contrast, we also show that the Euler characteristic in question can
take any integer value for a suitable graph. This disproves a conjecture of
Aluffi in a strong sense.
