{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,12,15]],"date-time":"2025-12-15T06:25:48Z","timestamp":1765779948449,"version":"3.48.0"},"reference-count":18,"publisher":"Springer Science and Business Media LLC","issue":"3","license":[{"start":{"date-parts":[[2025,12,15]],"date-time":"2025-12-15T00:00:00Z","timestamp":1765756800000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2025,12,15]],"date-time":"2025-12-15T00:00:00Z","timestamp":1765756800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"funder":[{"DOI":"10.13039\/501100012166","name":"National Key R&D Program of China","doi-asserted-by":"crossref","award":["2022YFA1006400"],"award-info":[{"award-number":["2022YFA1006400"]}],"id":[{"id":"10.13039\/501100012166","id-type":"DOI","asserted-by":"crossref"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["12571376"],"award-info":[{"award-number":["12571376"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100003395","name":"Shanghai Municipal Education Commission","doi-asserted-by":"publisher","award":["2024AIYB003"],"award-info":[{"award-number":["2024AIYB003"]}],"id":[{"id":"10.13039\/501100003395","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Comp. Appl. Math."],"published-print":{"date-parts":[[2026,4]]},"DOI":"10.1007\/s40314-025-03548-6","type":"journal-article","created":{"date-parts":[[2025,12,15]],"date-time":"2025-12-15T06:23:10Z","timestamp":1765779790000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Characterization of graphs with rank $$2v(G)-2\\alpha (G)-2c(G)+1$$"],"prefix":"10.1007","volume":"45","author":[{"given":"Feng","family":"Liu","sequence":"first","affiliation":[]},{"given":"Qi","family":"Wu","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2025,12,15]]},"reference":[{"key":"3548_CR1","doi-asserted-by":"publisher","first-page":"55","DOI":"10.1016\/S0024-3795(96)00513-7","volume":"265","author":"JH Bevis","year":"1997","unstructured":"Bevis JH, Blount KK, Davis GJ, Domke GS, Miller VA (1997) The rank of graph after vertex addition. Linear Algebra Appl 265:55\u201369","journal-title":"Linear Algebra Appl"},{"key":"3548_CR2","doi-asserted-by":"crossref","unstructured":"Bondy JA, Murty USR (2008) Graph theory. In: Axler S, Ribet KA (eds) Graduate texts in mathematics, vol 244. Springer, New York","DOI":"10.1007\/978-1-84628-970-5"},{"key":"3548_CR3","doi-asserted-by":"publisher","first-page":"60","DOI":"10.13001\/1081-3810.1182","volume":"16","author":"B Cheng","year":"2007","unstructured":"Cheng B, Liu BL (2007) On the nullity of graphs. Electron J Linear Algebra 16:60\u201367","journal-title":"Electron J Linear Algebra"},{"key":"3548_CR4","doi-asserted-by":"publisher","first-page":"192","DOI":"10.1016\/j.laa.2021.09.015","volume":"632","author":"B Cheng","year":"2022","unstructured":"Cheng B, Liu MH, Tam B-S (2022) On the nullity of a connected graph in terms of order and maximum degree. Linear Algebra Appl 632:192\u2013232","journal-title":"Linear Algebra Appl"},{"key":"3548_CR5","doi-asserted-by":"publisher","first-page":"63","DOI":"10.1007\/BF02941924","volume":"21","author":"L Collatz","year":"1957","unstructured":"Collatz L, Sinogowitz U (1957) Spektren endlicher grafen. Abh Math Sem Univ Hamburg 21:63\u201377","journal-title":"Abh Math Sem Univ Hamburg"},{"key":"3548_CR6","doi-asserted-by":"publisher","first-page":"158","DOI":"10.1016\/j.laa.2019.12.014","volume":"589","author":"SJ He","year":"2020","unstructured":"He SJ, Hao RX, Dong FM (2020) The rank of a complex unit gain graph in terms of the matching number. Linear Algebra Appl 589:158\u2013185","journal-title":"Linear Algebra Appl"},{"key":"3548_CR7","doi-asserted-by":"publisher","first-page":"65","DOI":"10.1007\/s10878-018-0282-x","volume":"36","author":"J Huang","year":"2018","unstructured":"Huang J, Li SC, Wang H (2018) Relation between the skew-rank of an oriented graph and the independence number of its underlying graph. J Comb Optim 36:65\u201380","journal-title":"J Comb Optim"},{"key":"3548_CR8","doi-asserted-by":"publisher","first-page":"130","DOI":"10.1016\/j.dam.2019.04.018","volume":"268","author":"X Li","year":"2019","unstructured":"Li X, Guo JM (2019) No graph with nullity $$\\eta (G)=|V(G)|-2m(G)+2c(G)-1$$. Discret Appl Math 268:130\u2013136","journal-title":"Discret Appl Math"},{"key":"3548_CR9","doi-asserted-by":"publisher","first-page":"2230","DOI":"10.1080\/03081087.2018.1488936","volume":"67","author":"SC Li","year":"2019","unstructured":"Li SC, Zhang SQ, Xu BG (2019) The relation between the $$H$$-rank of a mixed graph and the independence number of its underlying graph. Linear Multiliear Algebra 67:2230\u20132245","journal-title":"Linear Multiliear Algebra"},{"key":"3548_CR10","doi-asserted-by":"publisher","first-page":"171","DOI":"10.1016\/j.dam.2016.07.010","volume":"215","author":"XB Ma","year":"2016","unstructured":"Ma XB, Wong D, Tian FL (2016) Nullity of a graph in terms of the dimension of cycle space and the number of pendant vertices. Discret Appl Math 215:171\u2013176","journal-title":"Discret Appl Math"},{"key":"3548_CR11","doi-asserted-by":"publisher","first-page":"71","DOI":"10.1186\/s13660-016-1018-z","volume":"2016","author":"S Rula","year":"2016","unstructured":"Rula S, Chang A, Zheng YR (2016) The extremal graphs with respect to their nullity. J Inequal Appl 2016:71","journal-title":"J Inequal Appl"},{"key":"3548_CR12","doi-asserted-by":"publisher","first-page":"363","DOI":"10.1016\/j.laa.2014.09.034","volume":"465","author":"YZ Song","year":"2015","unstructured":"Song YZ, Song XQ, Tam BT (2015) A characterization of graphs $$G$$ with nullity $$|V(G)|-2m(G)+2c(G)$$. Linear Algebra Appl 465:363\u2013375","journal-title":"Linear Algebra Appl"},{"key":"3548_CR13","doi-asserted-by":"publisher","first-page":"276","DOI":"10.1016\/j.dam.2013.09.012","volume":"166","author":"L Wang","year":"2014","unstructured":"Wang L, Wong D (2014) Bounds for the matching number, the edge chromatic number and the independence number of a graph in terms of rank. Discret Appl Math 166:276\u2013281","journal-title":"Discret Appl Math"},{"key":"3548_CR14","doi-asserted-by":"publisher","first-page":"76","DOI":"10.1016\/j.ejc.2015.12.005","volume":"54","author":"D Wong","year":"2016","unstructured":"Wong D, Ma XB, Tian FL (2016) Relation between the skew-rank of an oriented graph and the rank of its underlying graph. Eur J Comb 54:76\u201386","journal-title":"Eur J Comb"},{"key":"3548_CR15","doi-asserted-by":"publisher","first-page":"166","DOI":"10.1016\/j.laa.2017.10.013","volume":"538","author":"L Yong","year":"2018","unstructured":"Yong L, Wang L, Zhou Q (2018) The rank of a signed graph in terms of the rank of its underlying graph. Linear Algebra Appl 538:166\u2013186","journal-title":"Linear Algebra Appl"},{"key":"3548_CR16","doi-asserted-by":"publisher","first-page":"314","DOI":"10.1016\/j.laa.2018.06.025","volume":"555","author":"Q Zhou","year":"2018","unstructured":"Zhou Q, Wong D, Sun DQ (2018) An upper bound of the nullity of a graph in terms of order and maximum degree. Linear Algebra Appl 555:314\u2013320","journal-title":"Linear Algebra Appl"},{"key":"3548_CR17","doi-asserted-by":"publisher","first-page":"56","DOI":"10.1016\/j.laa.2021.07.005","volume":"630","author":"Q Zhou","year":"2021","unstructured":"Zhou Q, Wong D, Tam BS (2021) On connected graphs of order $$n$$ with girth $$g$$ and nullity $$n-g$$. Linear Algebra Appl 630:56\u201368","journal-title":"Linear Algebra Appl"},{"key":"3548_CR18","doi-asserted-by":"publisher","first-page":"93","DOI":"10.1016\/j.dam.2022.01.020","volume":"313","author":"Q Zhou","year":"2022","unstructured":"Zhou Q, Wong D, Tian FL (2022) Relation between the nullity of a graph and its matching number. Discret Appl Math 313:93\u201398","journal-title":"Discret Appl Math"}],"container-title":["Computational and Applied Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s40314-025-03548-6.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s40314-025-03548-6\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s40314-025-03548-6.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,12,15]],"date-time":"2025-12-15T06:23:12Z","timestamp":1765779792000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s40314-025-03548-6"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,12,15]]},"references-count":18,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2026,4]]}},"alternative-id":["3548"],"URL":"https:\/\/doi.org\/10.1007\/s40314-025-03548-6","relation":{},"ISSN":["2238-3603","1807-0302"],"issn-type":[{"value":"2238-3603","type":"print"},{"value":"1807-0302","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,12,15]]},"assertion":[{"value":"7 January 2025","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"28 September 2025","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"17 November 2025","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"15 December 2025","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}}],"article-number":"111"}}