{"paper":{"title":"Spectral lower bounds for the orthogonal and projective ranks of a graph","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["quant-ph"],"primary_cat":"math.CO","authors_text":"Clive Elphick, Pawel Wocjan","submitted_at":"2018-06-07T15:40:14Z","abstract_excerpt":"The orthogonal rank of a graph $G=(V,E)$ is the smallest dimension $\\xi$ such that there exist non-zero column vectors $x_v\\in\\mathbb{C}^\\xi$ for $v\\in V$ satisfying the orthogonality condition $x_v^\\dagger x_w=0$ for all $vw\\in E$. We prove that many spectral lower bounds for the chromatic number, $\\chi$, are also lower bounds for $\\xi$. This result complements a previous result by the authors, in which they showed that spectral lower bounds for $\\chi$ are also lower bounds for the quantum chromatic number $\\chi_q$. It is known that the quantum chromatic number and the orthogonal rank are inc"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1806.02734","kind":"arxiv","version":3},"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"}