  We consider Artinian level algebras arising from the whiskering of a graph.
Employing a result by Dao-Nair we show that multiplication by a general linear
form has maximal rank in degrees 1 and $n-1$ when the characteristic is not
two, where $n$ is the number of vertices in the graph. Moreover, the
multiplication is injective in degrees $<n/2$ when the characteristic is zero,
following a proof by Hausel. Our result in the characteristic zero case is
optimal in the sense that there are whiskered graphs for which the
multiplication maps in all intermediate degrees $n/2,\ldots,n-2$ of the
associated Artinian algebras fail to have maximal rank, and consequently, the
weak Lefschetz property.
