The weak Lefschetz property of whiskered graphs

By employing a result by Dao--Nair, we show that multiplication by a general linear form in the Artinian algebra created from a whiskered graph with $n$ vertices always has maximal rank in degrees $1$ and $n-1$. This result is optimal in the sense that there are whiskered graphs whose associated Artinian algebras fail to have this property, and hence the weak Lefschetz property, in all intermediate degrees.