Graph hypersurfaces with torus action and a conjecture of Aluffi

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