{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,12,24]],"date-time":"2025-12-24T12:24:59Z","timestamp":1766579099503,"version":"3.41.2"},"reference-count":0,"publisher":"The Electronic Journal of Combinatorics","issue":"3","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Electron. J. Combin."],"abstract":"<jats:p>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 $vw\\in E$. \u00a0 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 $\\chi_q$.\u00a0 It is known that the quantum chromatic number and the orthogonal rank are incomparable. \u00a0\r\nWe conclude by proving an inertial\u00a0 lower bound for the projective rank $\\xi_f$, and conjecture that a stronger inertial lower bound for $\\xi$ is also a lower bound for $\\xi_f$.<\/jats:p>","DOI":"10.37236\/8183","type":"journal-article","created":{"date-parts":[[2020,1,10]],"date-time":"2020-01-10T01:45:40Z","timestamp":1578620740000},"source":"Crossref","is-referenced-by-count":3,"title":["Spectral Lower Bounds for the Orthogonal and Projective Ranks of a Graph"],"prefix":"10.37236","volume":"26","author":[{"given":"Pawel","family":"Wocjan","sequence":"first","affiliation":[]},{"given":"Clive","family":"Elphick","sequence":"additional","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2019,8,30]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v26i3p45\/7905","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v26i3p45\/7905","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,16]],"date-time":"2020-01-16T23:09:22Z","timestamp":1579216162000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v26i3p45"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,8,30]]},"references-count":0,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2019,7,4]]}},"URL":"https:\/\/doi.org\/10.37236\/8183","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2019,8,30]]},"article-number":"P3.45"}}