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Given a base field $\Bbbk$ of characteristic zero, for each graph $G$, weassociate the artinian algebra $A(G)$ defined by the edge ideal of $G$ and thesquares of the variables. We study the weak Lefschetz property of $A(G)$. Weclassify some classes of graphs with relatively few edges, including paths andcycles, such that its associated artinian ring has the weak Lefschetz property.

The weak Lefschetz property of artinian algebras associated to paths and cycles