{"paper":{"title":"A note on the values of independence polynomials at $-1$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Jonathan Cutler, Nathan Kahl","submitted_at":"2014-10-28T18:38:23Z","abstract_excerpt":"The independence polynomial $I(G;x)$ of a graph $G$ is $I(G;x)=\\sum_{k=1}^{\\alpha(G)} s_k x^k$, where $s_k$ is the number of independent sets in $G$ of size $k$. The decycling number of a graph $G$, denoted $\\phi(G)$, is the minimum size of a set $S\\subseteq V(G)$ such that $G-S$ is acyclic. Engstr\\\"om proved that the independence polynomial satisfies $|I(G;-1)| \\leq 2^{\\phi(G)}$ for any graph $G$, and this bound is best possible. Levit and Mandrescu provided an elementary proof of the bound, and in addition conjectured that for every positive integer $k$ and integer $q$ with $|q|\\leq 2^k$, th"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1410.7726","kind":"arxiv","version":1},"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"}