{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,19]],"date-time":"2026-03-19T21:34:43Z","timestamp":1773956083030,"version":"3.50.1"},"reference-count":27,"publisher":"Springer Science and Business Media LLC","issue":"3","license":[{"start":{"date-parts":[[2021,2,8]],"date-time":"2021-02-08T00:00:00Z","timestamp":1612742400000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"},{"start":{"date-parts":[[2021,2,8]],"date-time":"2021-02-08T00:00:00Z","timestamp":1612742400000},"content-version":"vor","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["11871398"],"award-info":[{"award-number":["11871398"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"name":"the Natural Science Basic Research Plan in Shaanxi Province of China","award":["Program No. 2018JM1032"],"award-info":[{"award-number":["Program No. 2018JM1032"]}]},{"name":"the Fundamental Research Funds for the Central Universities","award":["3102019ghjd003"],"award-info":[{"award-number":["3102019ghjd003"]}]},{"name":"the Innovation and Creation for Graduate Students in Northwestern Polytechnical University","award":["ZZ2019031"],"award-info":[{"award-number":["ZZ2019031"]}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["J Comb Optim"],"published-print":{"date-parts":[[2021,4]]},"DOI":"10.1007\/s10878-021-00704-6","type":"journal-article","created":{"date-parts":[[2021,2,9]],"date-time":"2021-02-09T07:16:55Z","timestamp":1612855015000},"page":"678-693","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["Some mixed graphs with H-rank 4,\u00a06 or 8"],"prefix":"10.1007","volume":"41","author":[{"given":"Jinling","family":"Yang","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6160-1761","authenticated-orcid":false,"given":"Ligong","family":"Wang","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Xiuwen","family":"Yang","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2021,2,8]]},"reference":[{"key":"704_CR1","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-84882-981-7","volume-title":"Graphs and matrices. Universitext","author":"RB Bapat","year":"2010","unstructured":"Bapat RB (2010) Graphs and matrices. Universitext. Springer, London"},{"key":"704_CR2","first-page":"23","volume":"25","author":"M Brown","year":"1993","unstructured":"Brown M, Kennedy JW, Servatius B (1993) Graph singularity. Graph Theory Notes N. Y. 25:23\u201332","journal-title":"Graph Theory Notes N. Y."},{"issue":"8","key":"704_CR3","doi-asserted-by":"publisher","first-page":"1793","DOI":"10.1016\/j.laa.2010.09.040","volume":"434","author":"GJ Chang","year":"2011","unstructured":"Chang GJ, Huang LH, Yeh HG (2011) A characterization of graphs with rank 4. Linear Algebra Appl 434(8):1793\u20131798","journal-title":"Linear Algebra Appl"},{"issue":"11","key":"704_CR4","doi-asserted-by":"publisher","first-page":"4241","DOI":"10.1016\/j.laa.2012.01.021","volume":"436","author":"GJ Chang","year":"2012","unstructured":"Chang GJ, Huang LH, Yeh HG (2012) A characterization of graphs with rank 5. Linear Algebra Appl 436(11):4241\u20134250","journal-title":"Linear Algebra Appl"},{"issue":"9","key":"704_CR5","doi-asserted-by":"publisher","first-page":"1853","DOI":"10.1080\/03081087.2017.1374327","volume":"66","author":"C Chen","year":"2018","unstructured":"Chen C, Huang J, Li SC (2018) On the relation between the $H$-rank of a mixed graph and the matching number of its underlying graph. Linear Multilinear Algebra 66(9):1853\u20131869","journal-title":"Linear Multilinear Algebra"},{"issue":"5","key":"704_CR6","doi-asserted-by":"publisher","first-page":"1300","DOI":"10.1016\/j.disc.2019.01.009","volume":"342","author":"C Chen","year":"2019","unstructured":"Chen C, Li SC, Zhang MJ (2019) Relation between the $H$-rank of a mixed graph and the rank of its underlying graph. Discrete Math 342(5):1300\u20131309","journal-title":"Discrete Math"},{"key":"704_CR7","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 dui Hamburg 21:63\u201377","journal-title":"Abh Math Sem Univ dui Hamburg"},{"key":"704_CR8","doi-asserted-by":"crossref","first-page":"411","DOI":"10.1016\/j.amc.2019.02.055","volume":"354","author":"ZM Feng","year":"2019","unstructured":"Feng ZM, Huang J, Li SC, Luo XB (2019) Relationship between the rank and the matching number of a graph. Appl Math Comput 354:411\u2013421","journal-title":"Appl Math Comput"},{"key":"704_CR9","doi-asserted-by":"publisher","first-page":"217","DOI":"10.1002\/jgt.22057","volume":"85","author":"K Guo","year":"2017","unstructured":"Guo K, Mohar B (2017a) Hermitian adjacency matrix of digraphs and mixed graphs. J Graph Theory 85:217\u2013248","journal-title":"J Graph Theory"},{"issue":"11","key":"704_CR10","doi-asserted-by":"publisher","first-page":"2616","DOI":"10.1016\/j.disc.2017.01.018","volume":"340","author":"K Guo","year":"2017","unstructured":"Guo K, Mohar B (2017b) Digraphs with Hermitian spectral radius below 2 and their cospectrality with paths. Discrete Math 340(11):2616\u20132632","journal-title":"Discrete Math"},{"key":"704_CR11","unstructured":"Gutman I, Borovi\u0107anin B (2011) Nullity of graphs: an updated survey. In: Selected topics on applications of graph spectra. Zb Rad (Beogr), vol 14, no 22, pp 137\u2013154"},{"key":"704_CR12","doi-asserted-by":"publisher","first-page":"273","DOI":"10.1016\/j.laa.2019.03.012","volume":"572","author":"SJ He","year":"2019","unstructured":"He SJ, Hao RX, Lai HJ (2019) Bounds for the matching number and cyclomatic number of a signed graph in terms of rank. Linear Algebra Appl 572:273\u2013291","journal-title":"Linear Algebra Appl"},{"issue":"1","key":"704_CR13","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(1):65\u201380","journal-title":"J Comb Optim"},{"issue":"1","key":"704_CR14","doi-asserted-by":"publisher","first-page":"268","DOI":"10.1007\/s10878-019-00382-5","volume":"38","author":"XL Li","year":"2019","unstructured":"Li XL, Xia W (2019) Skew-rank of an oriented graph and independence number of its underlying graph. J Comb Optim 38(1):268\u2013277","journal-title":"J Comb Optim"},{"key":"704_CR15","doi-asserted-by":"publisher","first-page":"182","DOI":"10.1016\/j.laa.2014.10.028","volume":"466","author":"JX Liu","year":"2015","unstructured":"Liu JX, Li XL (2015) Hermitian-adjacency matrices and Hermitian energies of mixed graphs. Linear Algebra Appl 466:182\u2013207","journal-title":"Linear Algebra Appl"},{"key":"704_CR16","doi-asserted-by":"crossref","first-page":"899","DOI":"10.1016\/j.amc.2015.08.105","volume":"270","author":"Y Lu","year":"2015","unstructured":"Lu Y, Wang LG, Zhou QN (2015) Bicyclic oriented graphs with skew-rank 6. Appl Math Comput 270:899\u2013908","journal-title":"Appl Math Comput"},{"key":"704_CR17","doi-asserted-by":"publisher","first-page":"169","DOI":"10.1016\/j.laa.2017.02.031","volume":"523","author":"Y Lu","year":"2017","unstructured":"Lu Y, Wang LG, Xiao P (2017) Complex unit gain bicyclic graphs with rank 2, 3 or 4. Linear Algebra Appl 523:169\u2013186","journal-title":"Linear Algebra Appl"},{"key":"704_CR18","doi-asserted-by":"publisher","first-page":"166","DOI":"10.1016\/j.laa.2017.10.013","volume":"538","author":"Y Lu","year":"2018","unstructured":"Lu Y, Wang LG, Zhou QN (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"},{"issue":"2","key":"704_CR19","doi-asserted-by":"publisher","first-page":"570","DOI":"10.1007\/s10878-019-00397-y","volume":"38","author":"Y Lu","year":"2019","unstructured":"Lu Y, Wang LG, Zhou QN (2019) The rank of a complex unit gain graph in terms of the rank of its underlying graph. J Comb Optim 38(2):570\u2013588","journal-title":"J Comb Optim"},{"key":"704_CR20","doi-asserted-by":"publisher","first-page":"205","DOI":"10.1016\/j.laa.2018.04.032","volume":"554","author":"WJ Luo","year":"2018","unstructured":"Luo WJ, Huang J, Li SC (2018) On the relationship between the skew-rank of an oriented graph and the rank of its underlying graph. Linear Algebra Appl 554:205\u2013223","journal-title":"Linear Algebra Appl"},{"key":"704_CR21","doi-asserted-by":"publisher","first-page":"324","DOI":"10.1016\/j.laa.2015.10.018","volume":"489","author":"B Mohar","year":"2016","unstructured":"Mohar B (2016) Hermitian adjacency spectrum and switching equivalence of mixed graphs. Linear Algebra Appl 489:324\u2013340","journal-title":"Linear Algebra Appl"},{"key":"704_CR22","doi-asserted-by":"crossref","first-page":"182","DOI":"10.1016\/j.amc.2015.02.014","volume":"258","author":"H Qu","year":"2015","unstructured":"Qu H, Yu GH (2015) Bicyclic oriented graphs with skew-rank 2 or 4. Appl Math Comput 258:182\u2013191","journal-title":"Appl Math Comput"},{"key":"704_CR23","doi-asserted-by":"publisher","first-page":"161","DOI":"10.1016\/j.laa.2007.07.002","volume":"427","author":"F Ros\u00e0rio","year":"2007","unstructured":"Ros\u00e0rio F, Cec\u00edlia P (2007) The minimum rank of matrices and the equivalence class graph. Linear Algebra Appl 427:161\u2013170","journal-title":"Linear Algebra Appl"},{"key":"704_CR24","doi-asserted-by":"publisher","first-page":"451","DOI":"10.13001\/1081-3810.1215","volume":"16","author":"I Sciriha","year":"2007","unstructured":"Sciriha I (2007) A characterization of singular graphs. Electron J Linear Algebra 16:451\u2013462","journal-title":"Electron J Linear Algebra"},{"key":"704_CR25","unstructured":"Wang CJ (2017) The nullity of unicyclic and bicyclic mixed graphs. Master thesis, Anhui University"},{"key":"704_CR26","doi-asserted-by":"publisher","first-page":"22","DOI":"10.1016\/j.laa.2017.02.037","volume":"524","author":"Y Wang","year":"2017","unstructured":"Wang Y, Yuan BJ, Li SD, Wang CJ (2017) Mixed graphs with $H$-rank 3. Linear Algebra Appl 524:22\u201334","journal-title":"Linear Algebra Appl"},{"key":"704_CR27","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 DI, 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"}],"container-title":["Journal of Combinatorial Optimization"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s10878-021-00704-6.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s10878-021-00704-6\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s10878-021-00704-6.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,10,30]],"date-time":"2023-10-30T01:44:32Z","timestamp":1698630272000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s10878-021-00704-6"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,2,8]]},"references-count":27,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2021,4]]}},"alternative-id":["704"],"URL":"https:\/\/doi.org\/10.1007\/s10878-021-00704-6","relation":{},"ISSN":["1382-6905","1573-2886"],"issn-type":[{"value":"1382-6905","type":"print"},{"value":"1573-2886","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,2,8]]},"assertion":[{"value":"19 January 2021","order":1,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"8 February 2021","order":2,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}