{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2022,3,31]],"date-time":"2022-03-31T15:14:37Z","timestamp":1648739677579},"reference-count":7,"publisher":"World Scientific Pub Co Pte Lt","issue":"02","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Discrete Math. Algorithm. Appl."],"published-print":{"date-parts":[[2009,6]]},"abstract":"<jats:p> O. Reingold et al. introduced the notion zig-zag product on two different graphs, and presented a fully explicit construction of d-regular expanders with the second largest eigenvalue O(d<jats:sup>-1\/3<\/jats:sup>). In the same paper, they ask whether or not the similar technique can be used to construct expanders with the second largest eigenvalue O(d<jats:sup>-1\/2<\/jats:sup>). Such graphs are called Ramanujan graphs. Recently, zig-zag product has been generalized by A. Ben-Aroya and A. Ta-Shma. Using this technique, they present a family of expanders with the second largest eigenvalue d<jats:sup>-1\/2 + o(1)<\/jats:sup>, what they call almost-Ramanujan graphs. However, their construction relies on local invertible functions and the dependence between the big graph and several small graphs, which makes the construction more complicated. <\/jats:p><jats:p> In this paper, we shall give a generalized theorem of zig-zag product. Specifically, the zig-zag product of one \"big\" graph and several \"small\" graphs with the same size will be formalized. By choosing the big graph and several small graphs individually, we shall present a family of fully explicitly almost-Ramanujan graphs with locally invertible function waived. <\/jats:p>","DOI":"10.1142\/s1793830909000154","type":"journal-article","created":{"date-parts":[[2009,7,2]],"date-time":"2009-07-02T11:53:30Z","timestamp":1246535610000},"page":"193-203","source":"Crossref","is-referenced-by-count":0,"title":["ON CONSTRUCTION OF ALMOST-RAMANUJAN GRAPHS"],"prefix":"10.1142","volume":"01","author":[{"given":"HE","family":"SUN","sequence":"first","affiliation":[{"name":"Shanghai Key Laboratory of Intelligent Information Processing, Fudan University, 220 Handan Road, Shanghai, 200433, P. R. China"}]},{"given":"HONG","family":"ZHU","sequence":"additional","affiliation":[{"name":"Shanghai Key Laboratory of Trustworthy Computing, East China Normal University, 3663 North Zhongshan Road, Shanghai, 200062, P. R. China"}]}],"member":"219","published-online":{"date-parts":[[2012,4,5]]},"reference":[{"key":"rf1","doi-asserted-by":"publisher","DOI":"10.1007\/BF01302963"},{"key":"rf5","doi-asserted-by":"publisher","DOI":"10.1007\/BF01285816"},{"key":"rf7","doi-asserted-by":"publisher","DOI":"10.1090\/S0273-0979-06-01126-8"},{"key":"rf8","doi-asserted-by":"publisher","DOI":"10.1007\/BF02126799"},{"key":"rf9","first-page":"71","volume":"9","author":"Margulis G. A.","journal-title":"Problemy Peredaci Informacii"},{"key":"rf10","doi-asserted-by":"publisher","DOI":"10.1006\/jctb.1994.1054"},{"key":"rf13","doi-asserted-by":"publisher","DOI":"10.1002\/1098-2418(200008)17:1<64::AID-RSA5>3.0.CO;2-3"}],"container-title":["Discrete Mathematics, Algorithms and Applications"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.worldscientific.com\/doi\/pdf\/10.1142\/S1793830909000154","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,8,7]],"date-time":"2019-08-07T15:57:53Z","timestamp":1565193473000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.worldscientific.com\/doi\/abs\/10.1142\/S1793830909000154"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2009,6]]},"references-count":7,"journal-issue":{"issue":"02","published-online":{"date-parts":[[2012,4,5]]},"published-print":{"date-parts":[[2009,6]]}},"alternative-id":["10.1142\/S1793830909000154"],"URL":"https:\/\/doi.org\/10.1142\/s1793830909000154","relation":{},"ISSN":["1793-8309","1793-8317"],"issn-type":[{"value":"1793-8309","type":"print"},{"value":"1793-8317","type":"electronic"}],"subject":[],"published":{"date-parts":[[2009,6]]}}}