{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,9,25]],"date-time":"2025-09-25T18:15:23Z","timestamp":1758824123747,"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 $G$ be a graph randomly selected from ${\\bf G}_{n, p}$, the space of Erd\u0151s-R\u00e9nyi Random graphs with parameters $n$ and $p$, where $p \\geq {\\log^6 n\\over n}$. Also, let $A$ be the adjacency matrix of $G$, and $v_1$ be the first eigenvector of $A$. We provide two short proofs of the following statement: For all $i \\in [n]$, for some constant $c&gt;0$ $$\\left|v_1(i) - {1\\over\\sqrt{n}}\\right| \\leq c {1\\over\\sqrt{n}} {\\log n\\over\\log (np)} \\sqrt{{\\log n\\over np}}$$ with probability $1 - o(1)$.  This gives nearly optimal bounds on the entrywise stability of the first eigenvector of (Erd\u0151s-R\u00e9nyi) Random graphs.  This question about entrywise bounds was motivated by a problem in unsupervised spectral clustering. We make some progress towards solving that problem.<\/jats:p>","DOI":"10.37236\/220","type":"journal-article","created":{"date-parts":[[2020,1,11]],"date-time":"2020-01-11T04:24:17Z","timestamp":1578716657000},"source":"Crossref","is-referenced-by-count":14,"title":["Entrywise Bounds for Eigenvectors of Random Graphs"],"prefix":"10.37236","volume":"16","author":[{"given":"Pradipta","family":"Mitra","sequence":"first","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2009,10,31]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v16i1r131\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v16i1r131\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,18]],"date-time":"2020-01-18T02:40:02Z","timestamp":1579315202000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v16i1r131"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2009,10,31]]},"references-count":0,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2009,1,7]]}},"URL":"https:\/\/doi.org\/10.37236\/220","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2009,10,31]]},"article-number":"R131"}}