{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T15:08:41Z","timestamp":1753888121317,"version":"3.41.2"},"reference-count":27,"publisher":"Wiley","issue":"1","license":[{"start":{"date-parts":[[2021,2,22]],"date-time":"2021-02-22T00:00:00Z","timestamp":1613952000000},"content-version":"vor","delay-in-days":52,"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":["11601002","11601001"],"award-info":[{"award-number":["11601002","11601001"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["onlinelibrary.wiley.com"],"crossmark-restriction":true},"short-container-title":["Complexity"],"published-print":{"date-parts":[[2021,1]]},"abstract":"<jats:p>Let <jats:italic>G<\/jats:italic> be a connected graph with minimum degree <jats:italic>\u03b4<\/jats:italic>(<jats:italic>G<\/jats:italic>) and vertex\u2010connectivity <jats:italic>\u03ba<\/jats:italic>(<jats:italic>G<\/jats:italic>). The graph <jats:italic>G<\/jats:italic> is <jats:italic>k<\/jats:italic>\u2010connected if <jats:italic>\u03ba<\/jats:italic>(<jats:italic>G<\/jats:italic>) \u2265 <jats:italic>k<\/jats:italic>, maximally connected if <jats:italic>\u03ba<\/jats:italic>(<jats:italic>G<\/jats:italic>) = <jats:italic>\u03b4<\/jats:italic>(<jats:italic>G<\/jats:italic>), and super\u2010connected if every minimum vertex\u2010cut isolates a vertex of minimum degree. In this paper, we present sufficient conditions for a graph with given minimum degree to be <jats:italic>k<\/jats:italic>\u2010connected, maximally connected, or super\u2010connected in terms of the number of edges, the spectral radius of the graph, and its complement, respectively. Analogous results for triangle\u2010free graphs with given minimum degree to be <jats:italic>k<\/jats:italic>\u2010connected, maximally connected, or super\u2010connected are also presented.<\/jats:p>","DOI":"10.1155\/2021\/5588146","type":"journal-article","created":{"date-parts":[[2021,2,23]],"date-time":"2021-02-23T03:05:17Z","timestamp":1614049517000},"update-policy":"https:\/\/doi.org\/10.1002\/crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Sufficient Conditions for Graphs to Be <i>k<\/i>\u2010Connected, Maximally Connected, and Super\u2010Connected"],"prefix":"10.1155","volume":"2021","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-1248-0227","authenticated-orcid":false,"given":"Zhen-Mu","family":"Hong","sequence":"first","affiliation":[]},{"given":"Zheng-Jiang","family":"Xia","sequence":"additional","affiliation":[]},{"given":"Fuyuan","family":"Chen","sequence":"additional","affiliation":[]},{"given":"Lutz","family":"Volkmann","sequence":"additional","affiliation":[]}],"member":"311","published-online":{"date-parts":[[2021,2,22]]},"reference":[{"volume-title":"Graph Theory. Graduate Texts in Mathematics","year":"2008","author":"Bondy J. A.","key":"e_1_2_9_1_2"},{"key":"e_1_2_9_2_2","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4419-8698-6"},{"key":"e_1_2_9_3_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.disc.2007.06.035"},{"key":"e_1_2_9_4_2","first-page":"35","article-title":"Sufficient conditions for maxmally edge-connected and super-edge-connected graphs","volume":"2","author":"Volkmann L.","year":"2017","journal-title":"Communications in Combinatorics and Optimization"},{"key":"e_1_2_9_5_2","doi-asserted-by":"publisher","DOI":"10.7151\/dmgt.2096"},{"key":"e_1_2_9_6_2","doi-asserted-by":"publisher","DOI":"10.21136\/cmj.1973.101168"},{"key":"e_1_2_9_7_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.laa.2009.01.005"},{"key":"e_1_2_9_8_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.laa.2010.01.007"},{"key":"e_1_2_9_9_2","doi-asserted-by":"publisher","DOI":"10.1080\/03081087.2018.1529732"},{"key":"e_1_2_9_10_2","doi-asserted-by":"publisher","DOI":"10.1002\/jgt.21857"},{"key":"e_1_2_9_11_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.laa.2013.10.017"},{"key":"e_1_2_9_12_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.laa.2019.05.022"},{"key":"e_1_2_9_13_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.laa.2014.09.015"},{"key":"e_1_2_9_14_2","first-page":"11","article-title":"Spectral conditions for a graph to be k-connected","volume":"8","author":"Li R.","year":"2014","journal-title":"Annals of Pure and Applied Mathematics"},{"key":"e_1_2_9_15_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.laa.2017.03.006"},{"key":"e_1_2_9_16_2","doi-asserted-by":"publisher","DOI":"10.1007\/s00605-017-1055-9"},{"key":"e_1_2_9_17_2","doi-asserted-by":"publisher","DOI":"10.13001\/1081-3810.3675"},{"key":"e_1_2_9_18_2","doi-asserted-by":"publisher","DOI":"10.1007\/s10587-016-0300-z"},{"key":"e_1_2_9_19_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.dam.2019.10.010"},{"key":"e_1_2_9_20_2","doi-asserted-by":"publisher","DOI":"10.1007\/s00373-019-02118-y"},{"key":"e_1_2_9_21_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.laa.2019.05.030"},{"key":"e_1_2_9_22_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.laa.2020.08.015"},{"key":"e_1_2_9_23_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.laa.2020.10.023"},{"key":"e_1_2_9_24_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.amc.2018.09.053"},{"key":"e_1_2_9_25_2","doi-asserted-by":"publisher","DOI":"10.1006\/jctb.2000.1997"},{"key":"e_1_2_9_26_2","first-page":"60","article-title":"Problem 28","volume":"10","author":"Mantel W.","year":"1907","journal-title":"Wiskundige Opgaven"},{"key":"e_1_2_9_27_2","first-page":"436","article-title":"On an extremal problem in graph theory (Hungarian)","volume":"48","author":"Tur\u00e1n P.","year":"1941","journal-title":"Mat Fiz. Lapok"}],"container-title":["Complexity"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/downloads.hindawi.com\/journals\/complexity\/2021\/5588146.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/downloads.hindawi.com\/journals\/complexity\/2021\/5588146.xml","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/pdf\/10.1155\/2021\/5588146","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,8,9]],"date-time":"2024-08-09T22:32:03Z","timestamp":1723242723000},"score":1,"resource":{"primary":{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/10.1155\/2021\/5588146"}},"subtitle":[],"editor":[{"given":"M. Irfan","family":"Uddin","sequence":"additional","affiliation":[]}],"short-title":[],"issued":{"date-parts":[[2021,1]]},"references-count":27,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2021,1]]}},"alternative-id":["10.1155\/2021\/5588146"],"URL":"https:\/\/doi.org\/10.1155\/2021\/5588146","archive":["Portico"],"relation":{},"ISSN":["1076-2787","1099-0526"],"issn-type":[{"type":"print","value":"1076-2787"},{"type":"electronic","value":"1099-0526"}],"subject":[],"published":{"date-parts":[[2021,1]]},"assertion":[{"value":"2021-01-07","order":0,"name":"received","label":"Received","group":{"name":"publication_history","label":"Publication History"}},{"value":"2021-01-27","order":2,"name":"accepted","label":"Accepted","group":{"name":"publication_history","label":"Publication History"}},{"value":"2021-02-22","order":3,"name":"published","label":"Published","group":{"name":"publication_history","label":"Publication History"}}],"article-number":"5588146"}}