{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,27]],"date-time":"2025-11-27T20:56:33Z","timestamp":1764276993915,"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>We give a simple proof of the Alon\u2013Roichman theorem, which asserts that the Cayley graph obtained by selecting $c_\\varepsilon \\log |G|$ elements, independently and uniformly at random, from a finite group $G$ has expected second eigenvalue no more than $\\varepsilon$; here $c_\\varepsilon$ is a constant that depends only on $\\varepsilon$. In particular, such a graph is an expander with constant probability. Our new proof has three advantages over the original proof: (i.) it is extremely simple, relying only on the decomposition of the group algebra and tail bounds for operator-valued random variables, (ii.) it shows that the $\\log |G|$ term may be replaced with $\\log D$, where $D \\leq |G|$ is the sum of the dimensions of the irreducible representations of $G$, and (iii.) it establishes the result above with a smaller constant $c_\\varepsilon$.<\/jats:p>","DOI":"10.37236\/1815","type":"journal-article","created":{"date-parts":[[2020,1,10]],"date-time":"2020-01-10T21:32:09Z","timestamp":1578691929000},"source":"Crossref","is-referenced-by-count":16,"title":["Random Cayley Graphs are Expanders: a Simple Proof of the  Alon&amp;ndash;Roichman Theorem"],"prefix":"10.37236","volume":"11","author":[{"given":"Zeph","family":"Landau","sequence":"first","affiliation":[]},{"given":"Alexander","family":"Russell","sequence":"additional","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2004,9,13]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v11i1r62\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v11i1r62\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,17]],"date-time":"2020-01-17T23:54:05Z","timestamp":1579305245000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v11i1r62"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2004,9,13]]},"references-count":0,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2004,1,2]]}},"URL":"https:\/\/doi.org\/10.37236\/1815","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2004,9,13]]},"article-number":"R62"}}