The Weak Lefschetz Property of Artinian Algebras Associated to Paths and Cycles

Given a base field k\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Bbbk $$\end{document} of characteristic zero, for each graph G, we associate the artinian algebra A(G) defined by the edge ideal of G and the squares of the variables. We study the weak Lefschetz property of A(G). We classify some classes of graphs with relatively few edges, including paths and cycles, such that its associated artinian ring has the weak Lefschetz property.