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/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,..., n - 2 of the associated Artinian algebras fail to have maximal rank, and consequently, the weak Lefschetz property.