{"paper":{"title":"Unimodality of the independence polynomials of some composite graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Bao-Xuan Zhu, Qinglin Lu","submitted_at":"2015-07-21T09:11:15Z","abstract_excerpt":"Let $I(G;x)$ denote the independence polynomial of a graph $G$. In this paper we study the unimodality properties of $I(G;x)$ for some composite graphs $G$.\n  Given two graphs $G_1$ and $G_2$, let $G_1[G_2]$ denote the lexicographic product of $G_1$ and $G_2$. Assume $I(G_1;x)=\\sum_{i\\geq0}a_ix^i$ and $I(G_2;x)=\\sum_{i\\geq0}b_ix^i$, where $I(G_2;x)$ is log-concave. Then we prove (i) if $I(G_1;x)$ is log-concave and $(a^2_i-a_{i-1}a_{i+1})b^2_1\\geq a_ia_{i-1}b_2$ for all $1\\leq i \\leq \\alpha(G_1)$, then $I(G_1[G_2];x)$ is log-concave; (ii) if $a_{i-1}\\leq b_1a_i$ for $1\\leq i\\leq \\alpha(G_1)$, "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1507.05754","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"}