{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,12,30]],"date-time":"2025-12-30T11:16:50Z","timestamp":1767093410157,"version":"3.41.2"},"reference-count":38,"publisher":"Wiley","issue":"1","license":[{"start":{"date-parts":[[2019,4,16]],"date-time":"2019-04-16T00:00:00Z","timestamp":1555372800000},"content-version":"vor","delay-in-days":105,"URL":"http:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["11601511"],"award-info":[{"award-number":["11601511"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100004608","name":"Natural Science Foundation of Jiangsu Province","doi-asserted-by":"publisher","award":["BK20160237","BK20150169"],"award-info":[{"award-number":["BK20160237","BK20150169"]}],"id":[{"id":"10.13039\/501100004608","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["onlinelibrary.wiley.com"],"crossmark-restriction":true},"short-container-title":["Complexity"],"published-print":{"date-parts":[[2019,1]]},"abstract":"<jats:p>The matching energy <jats:italic>M<\/jats:italic><jats:italic>E<\/jats:italic>(<jats:italic>G<\/jats:italic>) of a graph <jats:italic>G<\/jats:italic> was introduced by Gutman and Wagner, which is defined as the sum of the absolute values of the roots of the matching polynomial <jats:italic>m<\/jats:italic>(<jats:italic>G<\/jats:italic>, <jats:italic>x<\/jats:italic>). The largest matching root <jats:italic>\u03bb<\/jats:italic><jats:sub>1<\/jats:sub>(<jats:italic>G<\/jats:italic>) is the largest root of the matching polynomial <jats:italic>m<\/jats:italic>(<jats:italic>G<\/jats:italic>, <jats:italic>x<\/jats:italic>). Let  denote the complete <jats:italic>r<\/jats:italic>\u2010partite graph with order <jats:italic>n<\/jats:italic> = <jats:italic>n<\/jats:italic><jats:sub>1<\/jats:sub> + <jats:italic>n<\/jats:italic><jats:sub>2<\/jats:sub> + \u2026+<jats:italic>n<\/jats:italic><jats:sub><jats:italic>r<\/jats:italic><\/jats:sub>, where <jats:italic>r<\/jats:italic> &gt; 1. In this paper, we prove that, for the given values <jats:italic>n<\/jats:italic> and <jats:italic>r<\/jats:italic>, both the matching energy <jats:italic>M<\/jats:italic><jats:italic>E<\/jats:italic>(<jats:italic>G<\/jats:italic>) and the largest matching root <jats:italic>\u03bb<\/jats:italic><jats:sub>1<\/jats:sub>(<jats:italic>G<\/jats:italic>) of complete <jats:italic>r<\/jats:italic>\u2010partite graphs are minimal for complete split graph <jats:italic>C<\/jats:italic><jats:italic>S<\/jats:italic>(<jats:italic>n<\/jats:italic>, <jats:italic>r<\/jats:italic> \u2212 1) and are maximal for Tur\u00e1n graph <jats:italic>T<\/jats:italic>(<jats:italic>n<\/jats:italic>, <jats:italic>r<\/jats:italic>).<\/jats:p>","DOI":"10.1155\/2019\/9728976","type":"journal-article","created":{"date-parts":[[2019,4,17]],"date-time":"2019-04-17T00:53:35Z","timestamp":1555462415000},"update-policy":"https:\/\/doi.org\/10.1002\/crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Extremal Matching Energy and the Largest Matching Root of Complete Multipartite Graphs"],"prefix":"10.1155","volume":"2019","author":[{"given":"Xiaolin","family":"Chen","sequence":"first","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4208-3927","authenticated-orcid":false,"given":"Huishu","family":"Lian","sequence":"additional","affiliation":[]}],"member":"311","published-online":{"date-parts":[[2019,4,16]]},"reference":[{"key":"e_1_2_8_1_2","doi-asserted-by":"publisher","DOI":"10.1016\/0095-8956(79)90070-4"},{"key":"e_1_2_8_2_2","doi-asserted-by":"publisher","DOI":"10.1002\/jgt.3190050203"},{"key":"e_1_2_8_3_2","first-page":"75","article-title":"The matching polynomial","volume":"6","author":"Gutman I.","year":"1979","journal-title":"MATCH Communications in Mathematical and in Computer Chemistry"},{"key":"e_1_2_8_4_2","doi-asserted-by":"publisher","DOI":"10.1007\/BF01877590"},{"volume-title":"Algebraic Combinatorics","year":"1993","author":"Godsil C.","key":"e_1_2_8_5_2"},{"key":"e_1_2_8_6_2","doi-asserted-by":"publisher","DOI":"10.1016\/0012-365X(92)90718-U"},{"key":"e_1_2_8_7_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.dam.2017.02.009"},{"key":"e_1_2_8_8_2","doi-asserted-by":"publisher","DOI":"10.1007\/s00373-012-1186-7"},{"key":"e_1_2_8_9_2","first-page":"73","volume-title":"Graph Polynomials","author":"Gutman I.","year":"2016"},{"key":"e_1_2_8_10_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.dam.2016.12.022"},{"key":"e_1_2_8_11_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.ipl.2013.07.003"},{"key":"e_1_2_8_12_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.ipl.2014.10.012"},{"key":"e_1_2_8_13_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.dam.2012.06.001"},{"key":"e_1_2_8_14_2","first-page":"1","article-title":"The energy of a graph","volume":"103","author":"Gutman I.","year":"1978","journal-title":"Berichte der Mathematisch-Statistischen Sektion im Forschungszentrum Graz"},{"key":"e_1_2_8_15_2","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4614-4220-2"},{"key":"e_1_2_8_16_2","doi-asserted-by":"publisher","DOI":"10.1007\/BF00552542"},{"key":"e_1_2_8_17_2","doi-asserted-by":"publisher","DOI":"10.1021\/ja00448a002"},{"key":"e_1_2_8_18_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.amc.2015.06.115"},{"key":"e_1_2_8_19_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.amc.2015.10.025"},{"key":"e_1_2_8_20_2","doi-asserted-by":"publisher","DOI":"10.1002\/cplx.21599"},{"key":"e_1_2_8_21_2","first-page":"315","article-title":"Bounds on the matching energy of unicyclic odd-cycle graphs","volume":"75","author":"Chen L.","year":"2016","journal-title":"MATCH - 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