{"paper":{"title":"Critical ideals of trees","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AC"],"primary_cat":"math.CO","authors_text":"Carlos E. 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 $\\mathcal{M}$. We give a combinatorial description of a set of generators of the"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1504.06239","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}