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Valencia, Hugo Corrales","submitted_at":"2015-04-23T16:11:41Z","abstract_excerpt":"Given a graph $G=(V, E)$, its generalized Laplacian matrix is given by \\[ L(G,X_G)_{u,v}= \\begin{cases} x_u&\\text{if }u=v,\\\\ -m_{uv}&\\text{if }u\\neq v, \\end{cases} \\] where $X_G=\\{x_u\\, | \\, u\\in V(G)\\}$ is a set of indeterminates and $m_{uv}$ is the number of edges between $u$ and $v$. The $j$-critical ideal of $G$ is the determinantal ideal generated by the minors of size $j$ of $L(G, X)$. A $2$-matching of $G$ is a subset $\\mathcal{M}$ of its edges such that every vertex of $G$ has at most two incident edges in $\\mathcal{M}$. 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