{"paper":{"title":"Bounds on Characteristic Polynomials","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Fengwei Zhou, Suijie Wang, Yeong-Nan Yeh","submitted_at":"2012-09-24T08:17:05Z","abstract_excerpt":"Suppose $G$ is a simple graph with $n$ vertices, $m$ edges, and rank $r$. Let $\\chi_G(t)=a_0t^n-a_1t^{n-1}+\\cdots +(-1)^ra_rt^{n-r}$ be the chromatic polynomial of $G$. For $q,k\\in \\Bbb{Z}$ and $0\\le k\\le q+r+1$, we obtain a sharp two-side bound for the partial binomial sum of the coefficient sequence, that is, \\[ {r+q\\choose k}\\le \\sum_{i=0}^{k}{q\\choose k-i}a_{i}\\le {m+q\\choose k}. \\] Indeed, this bound holds for the characteristic polynomial of hyperplane arrangements and matroids, and its weak version can be generalized to the characteristic polynomial of toric arrangements and arithmetic "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1209.5185","kind":"arxiv","version":4},"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"}