The weak Lefschetz property of whiskered graphs

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 $<n\mathrm{/}2$ 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 $n\mathrm{/}2,\dots ,n-2$ of the associated Artinian algebras fail to have maximal rank, and consequently, the weak Lefschetz property.