{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2024,8,2]],"date-time":"2024-08-02T02:10:08Z","timestamp":1722564608459},"reference-count":10,"publisher":"Wiley","issue":"4","license":[{"start":{"date-parts":[[2006,10,5]],"date-time":"2006-10-05T00:00:00Z","timestamp":1160006400000},"content-version":"vor","delay-in-days":6243,"URL":"http:\/\/onlinelibrary.wiley.com\/termsAndConditions#vor"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Journal of Graph Theory"],"published-print":{"date-parts":[[1989,9]]},"abstract":"Abstract<\/jats:title>For a simple graph G<\/jats:italic>, let f<\/jats:italic>(z<\/jats:italic>) \u2261 1 \u2212 c<\/jats:italic>1<\/jats:sub>z<\/jats:italic> + c<\/jats:italic>3<\/jats:sub>z<\/jats:italic>2<\/jats:sup> \u2212 c<\/jats:italic>3<\/jats:sub>z<\/jats:italic>3<\/jats:sup> + \u2026 where c<\/jats:italic>k<\/jats:italic><\/jats:sub> is the number of complete subgraphs on k<\/jats:italic> nodes in G<\/jats:italic>. Let r<\/jats:italic>(G<\/jats:italic>) be the reciprocal of the smallest real root of f<\/jats:italic>(z<\/jats:italic>). Let \u03bb(\u1e20<\/jats:italic>) be the spectral radius of the complement of G<\/jats:italic>. We show r<\/jats:italic>(G<\/jats:italic>) \u2a7e \u03bb(\u1e20<\/jats:italic>) + 1. This is used to show that if c<\/jats:italic>2<\/jats:sup>1<\/jats:sub>\/4 \u2a7d c<\/jats:italic>2<\/jats:sub> \u2a7d c<\/jats:italic>2<\/jats:sup>1<\/jats:sub>\/3, then a lower bound on the number of triangles in G<\/jats:italic> is c<\/jats:italic>3<\/jats:sub> \u2a7e [9c<\/jats:italic>2<\/jats:sub>c<\/jats:italic>1<\/jats:sub> \u2212 2c<\/jats:italic>2<\/jats:sup>1<\/jats:sub> \u2212 2(c<\/jats:italic>2<\/jats:sup>1<\/jats:sub> \u2212 3c<\/jats:italic>2<\/jats:sub>)3\/2<\/jats:sup>]\/27. This improves a bound of Bollob\u00e1s and is asymptotically sharp.<\/jats:p>Also, this paper shows that magnified image<\/jats:alt-text><\/jats:graphic><\/jats:chem-struct><\/jats:chem-struct-wrap> (a corollary of a result from Erd\u00f6s and Hanini) and the average number of triangles in a graph with c<\/jats:italic>1<\/jats:sub> nodes and c<\/jats:italic>2<\/jats:sub> edges is E<\/jats:italic>(c<\/jats:italic>3<\/jats:sub>) = 3\/4(C<\/jats:italic>2<\/jats:sup>2<\/jats:sub> \u2212 3c<\/jats:italic>2<\/jats:sub> + 2)c<\/jats:italic>2<\/jats:sub>\/(c<\/jats:italic>3<\/jats:sup>1<\/jats:sub> \u2212 5c<\/jats:italic>1<\/jats:sub> \u2212 4). These are graphically compared to the best known lower bounds.<\/jats:p>","DOI":"10.1002\/jgt.3190130411","type":"journal-article","created":{"date-parts":[[2007,5,26]],"date-time":"2007-05-26T12:20:30Z","timestamp":1180182030000},"page":"505-512","source":"Crossref","is-referenced-by-count":41,"title":["Lower bounds on the number of triangles in a graph"],"prefix":"10.1002","volume":"13","author":[{"given":"David C.","family":"Fisher","sequence":"first","affiliation":[]}],"member":"311","published-online":{"date-parts":[[2006,10,5]]},"reference":[{"key":"e_1_2_1_2_2","volume-title":"Extremal Graph Theory","author":"Bollob\u00e1s B.","year":"1978"},{"key":"e_1_2_1_3_2","first-page":"459","article-title":"On the number of complete subgraphs contained in certain graphs","volume":"7","author":"Erd\u00f6s P.","year":"1962","journal-title":"Pub. Math. Institute Hung. Acad. Sci."},{"key":"e_1_2_1_4_2","doi-asserted-by":"publisher","DOI":"10.1016\/0012-365X(88)90021-0"},{"key":"e_1_2_1_5_2","article-title":"The number of words of length n in a free \u201csemi\u2010abelian\u201d monoid","author":"Fisher D. C.","journal-title":"Am. Math. Month."},{"key":"e_1_2_1_6_2","article-title":"Dependence polynomials","author":"Fisher D. C.","journal-title":"Discrete Math."},{"key":"e_1_2_1_7_2","doi-asserted-by":"crossref","unstructured":"L.Lov\u00e1szandM.Simonovits On complete subgraphs of a graph II.Studies Pure Math.(1983)459\u2013496.","DOI":"10.1007\/978-3-0348-5438-2_41"},{"key":"e_1_2_1_8_2","first-page":"969","article-title":"Solution of the problem of P. Erd\u00f6s on the number of triangles in graphs with n vertices and [n\n 2\/4] + l edges","volume":"34","author":"Nikiforov V. S.","year":"1981","journal-title":"C.R. Acad. Bulg. Sci."},{"key":"e_1_2_1_9_2","doi-asserted-by":"publisher","DOI":"10.4153\/CJM-1963-004-7"},{"key":"e_1_2_1_10_2","volume-title":"Graphical Evolution","author":"Palmer E. M.","year":"1985"},{"key":"e_1_2_1_11_2","first-page":"307","volume-title":"Selected Topics in Graph Theory","author":"Schwenk A. J.","year":"1978"}],"container-title":["Journal of Graph Theory"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/api.wiley.com\/onlinelibrary\/tdm\/v1\/articles\/10.1002%2Fjgt.3190130411","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/pdf\/10.1002\/jgt.3190130411","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,10,22]],"date-time":"2023-10-22T14:03:39Z","timestamp":1697983419000},"score":1,"resource":{"primary":{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/10.1002\/jgt.3190130411"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1989,9]]},"references-count":10,"journal-issue":{"issue":"4","published-print":{"date-parts":[[1989,9]]}},"alternative-id":["10.1002\/jgt.3190130411"],"URL":"https:\/\/doi.org\/10.1002\/jgt.3190130411","archive":["Portico"],"relation":{},"ISSN":["0364-9024","1097-0118"],"issn-type":[{"value":"0364-9024","type":"print"},{"value":"1097-0118","type":"electronic"}],"subject":[],"published":{"date-parts":[[1989,9]]}}}