{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T16:43:23Z","timestamp":1753893803706,"version":"3.41.2"},"reference-count":0,"publisher":"The Electronic Journal of Combinatorics","issue":"1","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Electron. J. Combin."],"abstract":"<jats:p>Let $$ {\\cal B}(G) =\\{X : X \\in{\\Bbb R}^{n \\times n}, X=X^T, I \\le X \\le I+A(G)\\} $$ and $$ {\\cal C}(G) =\\{X : X \\in{\\Bbb R}^{n \\times n}, X=X^T,  I-A(G) \\le X \\le I+A(G)\\} $$ be classes of matrices associated with graph $G$. Here $n$ is the number of vertices in graph $G$, and $A(G)$ is the adjacency matrix of this graph. Denote $r(G)=\\min_{X \\in {\\cal C}(G)} {\\rm rank}(X)$, $r_+(G)=\\min_{X \\in {\\cal B}(G)} {\\rm rank}(X)$. We have shown previously that for every graph $G$, $\\alpha(G)  \\le r_+(G) \\le \\overline \\chi(G)$ holds and $\\alpha(G)=r_+(G)$ implies $\\alpha(G)=\\overline \\chi(G)$. In this article we show that there is a graph $G$ such that $\\alpha(G)=r(G)$ but  $ \\alpha(G)  &lt;  \\overline \\chi(G).$ In the case when the graph $G$ doesn't contain two chordless cycles $C_4$ with a common edge, the equality  $\\alpha(G)=r(G)$ implies $ \\alpha(G) = \\overline \\chi(G)$. Corollary: the last statement holds for  $d(G)$ \u2013 the minimal dimension of the orthonormal representation of the graph $G$.<\/jats:p>","DOI":"10.37236\/1846","type":"journal-article","created":{"date-parts":[[2020,1,11]],"date-time":"2020-01-11T05:22:54Z","timestamp":1578720174000},"source":"Crossref","is-referenced-by-count":0,"title":["On the Functions with Values in $[\\alpha(G), \\overline \\chi(G)]$"],"prefix":"10.37236","volume":"11","author":[{"given":"V.","family":"Dobrynin","sequence":"first","affiliation":[]},{"given":"M.","family":"Pliskin","sequence":"additional","affiliation":[]},{"given":"E.","family":"Prosolupov","sequence":"additional","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2004,3,22]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v11i1n5\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v11i1n5\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,18]],"date-time":"2020-01-18T05:04:15Z","timestamp":1579323855000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v11i1n5"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2004,3,22]]},"references-count":0,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2004,1,2]]}},"URL":"https:\/\/doi.org\/10.37236\/1846","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2004,3,22]]},"article-number":"N5"}}