{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,19]],"date-time":"2026-03-19T13:19:55Z","timestamp":1773926395201,"version":"3.50.1"},"reference-count":32,"publisher":"Elsevier BV","issue":"8","license":[{"start":{"date-parts":[[2026,8,1]],"date-time":"2026-08-01T00:00:00Z","timestamp":1785542400000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.elsevier.com\/tdm\/userlicense\/1.0\/"},{"start":{"date-parts":[[2026,8,1]],"date-time":"2026-08-01T00:00:00Z","timestamp":1785542400000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.elsevier.com\/legal\/tdmrep-license"},{"start":{"date-parts":[[2026,8,1]],"date-time":"2026-08-01T00:00:00Z","timestamp":1785542400000},"content-version":"stm-asf","delay-in-days":0,"URL":"https:\/\/doi.org\/10.15223\/policy-017"},{"start":{"date-parts":[[2026,8,1]],"date-time":"2026-08-01T00:00:00Z","timestamp":1785542400000},"content-version":"stm-asf","delay-in-days":0,"URL":"https:\/\/doi.org\/10.15223\/policy-037"},{"start":{"date-parts":[[2026,8,1]],"date-time":"2026-08-01T00:00:00Z","timestamp":1785542400000},"content-version":"stm-asf","delay-in-days":0,"URL":"https:\/\/doi.org\/10.15223\/policy-012"},{"start":{"date-parts":[[2026,8,1]],"date-time":"2026-08-01T00:00:00Z","timestamp":1785542400000},"content-version":"stm-asf","delay-in-days":0,"URL":"https:\/\/doi.org\/10.15223\/policy-029"},{"start":{"date-parts":[[2026,8,1]],"date-time":"2026-08-01T00:00:00Z","timestamp":1785542400000},"content-version":"stm-asf","delay-in-days":0,"URL":"https:\/\/doi.org\/10.15223\/policy-004"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["elsevier.com","sciencedirect.com"],"crossmark-restriction":true},"short-container-title":["Discrete Mathematics"],"published-print":{"date-parts":[[2026,8]]},"DOI":"10.1016\/j.disc.2026.115119","type":"journal-article","created":{"date-parts":[[2026,3,19]],"date-time":"2026-03-19T07:49:53Z","timestamp":1773906593000},"page":"115119","update-policy":"https:\/\/doi.org\/10.1016\/elsevier_cm_policy","source":"Crossref","is-referenced-by-count":0,"title":["Improved bounds on the H-rank of a mixed graph in terms of the matching number and fractional matching number"],"prefix":"10.1016","volume":"349","author":[{"given":"Qi","family":"Wu","sequence":"first","affiliation":[]},{"given":"Yong","family":"Lu","sequence":"additional","affiliation":[]}],"member":"78","reference":[{"key":"10.1016\/j.disc.2026.115119_br0010","series-title":"Physical Chemistry","author":"Atkins","year":"2006"},{"key":"10.1016\/j.disc.2026.115119_br0020","doi-asserted-by":"crossref","first-page":"55","DOI":"10.1016\/S0024-3795(96)00513-7","article-title":"The rank of graph after vertex addition","volume":"265","author":"Bevis","year":"1997","journal-title":"Linear Algebra Appl."},{"key":"10.1016\/j.disc.2026.115119_br0030","series-title":"Graph Theory with Applications","author":"Bondy","year":"1976"},{"key":"10.1016\/j.disc.2026.115119_br0040","doi-asserted-by":"crossref","first-page":"1853","DOI":"10.1080\/03081087.2017.1374327","article-title":"On the relation between the H-rank of a mixed graph and the matching number of its underlying graph","volume":"66","author":"Chen","year":"2018","journal-title":"Linear Multilinear Algebra"},{"key":"10.1016\/j.disc.2026.115119_br0050","doi-asserted-by":"crossref","first-page":"1300","DOI":"10.1016\/j.disc.2019.01.009","article-title":"Relation between the H-rank of a mixed graph and the rank of its underlying graph","volume":"342","author":"Chen","year":"2019","journal-title":"Discrete Math."},{"key":"10.1016\/j.disc.2026.115119_br0060","doi-asserted-by":"crossref","DOI":"10.1016\/j.disc.2022.112919","article-title":"The nullity of a graph with fractional matching number","volume":"345","author":"Chen","year":"2022","journal-title":"Discrete Math."},{"key":"10.1016\/j.disc.2026.115119_br0070","doi-asserted-by":"crossref","first-page":"407","DOI":"10.1016\/j.laa.2022.06.024","article-title":"The rank of a signed graph","volume":"651","author":"Chen","year":"2022","journal-title":"Linear Algebra Appl."},{"key":"10.1016\/j.disc.2026.115119_br0080","doi-asserted-by":"crossref","first-page":"569","DOI":"10.1016\/j.laa.2024.07.006","article-title":"Bounds of nullity for complex unit gain graphs","volume":"699","author":"Chen","year":"2024","journal-title":"Linear Algebra Appl."},{"key":"10.1016\/j.disc.2026.115119_br0090","doi-asserted-by":"crossref","first-page":"60","DOI":"10.13001\/1081-3810.1182","article-title":"On the nullity of graphs","volume":"16","author":"Cheng","year":"2007","journal-title":"Electron. J. Linear Algebra"},{"key":"10.1016\/j.disc.2026.115119_br0100","doi-asserted-by":"crossref","first-page":"63","DOI":"10.1007\/BF02941924","article-title":"Spektren endlicher grafen","volume":"21","author":"Collatz","year":"1957","journal-title":"Abh. Math. Semin. Univ. Hamb."},{"key":"10.1016\/j.disc.2026.115119_br0110","first-page":"141","article-title":"The algebraic multiplicity of the number zero in the spectrum of a bipartite graph","volume":"9","author":"Cvetkovi\u0107","year":"1972","journal-title":"Mat. Vesn. (Beograd)"},{"key":"10.1016\/j.disc.2026.115119_br0130","first-page":"411","article-title":"Relationship between the rank and the matching number of a graph","volume":"354","author":"Feng","year":"2019","journal-title":"Appl. Math. Comput."},{"key":"10.1016\/j.disc.2026.115119_br0150","doi-asserted-by":"crossref","first-page":"1293","DOI":"10.1016\/j.laa.2009.04.026","article-title":"On the nullity and matching number of unicyclic graphs","volume":"431","author":"Guo","year":"2009","journal-title":"Linear Algebra Appl."},{"key":"10.1016\/j.disc.2026.115119_br0140","doi-asserted-by":"crossref","first-page":"273","DOI":"10.1016\/j.laa.2019.03.012","article-title":"Bounds for the matching number and cyclomatic number of a signed graph in terms of rank","volume":"572","author":"He","year":"2019","journal-title":"Linear Algebra Appl."},{"key":"10.1016\/j.disc.2026.115119_br0160","author":"He"},{"key":"10.1016\/j.disc.2026.115119_br0170","doi-asserted-by":"crossref","first-page":"239","DOI":"10.1016\/j.dam.2025.05.006","article-title":"Relation between the H-rank of a mixed graph and the girth of its underlying graph","volume":"373","author":"Khan","year":"2025","journal-title":"Discrete Appl. Math."},{"key":"10.1016\/j.disc.2026.115119_br0180","doi-asserted-by":"crossref","first-page":"2230","DOI":"10.1080\/03081087.2018.1488936","article-title":"The relation between the H-rank of a mixed graph and the independence number of its underlying graph","volume":"67","author":"Li","year":"2019","journal-title":"Linear Multilinear Algebra"},{"key":"10.1016\/j.disc.2026.115119_br0190","doi-asserted-by":"crossref","first-page":"130","DOI":"10.1016\/j.dam.2019.04.018","article-title":"No graph with nullity \u03b7(G)=|V(G)|\u22122m(G)+2\u03bc(G)\u22121","volume":"268","author":"Li","year":"2019","journal-title":"Discrete Appl. Math."},{"key":"10.1016\/j.disc.2026.115119_br0200","doi-asserted-by":"crossref","first-page":"169","DOI":"10.1016\/j.laa.2017.02.031","article-title":"Complex unit gain bicyclic graphs with rank 2, 3 or 4","volume":"523","author":"Lu","year":"2017","journal-title":"Linear Algebra Appl."},{"key":"10.1016\/j.disc.2026.115119_br0210","doi-asserted-by":"crossref","first-page":"1983","DOI":"10.1080\/03081087.2019.1567672","article-title":"An improved lower bound for the nullity of a graph in terms of matching number","volume":"68","author":"Ma","year":"2020","journal-title":"Linear Multilinear Algebra"},{"key":"10.1016\/j.disc.2026.115119_br0220","doi-asserted-by":"crossref","first-page":"171","DOI":"10.1016\/j.dam.2016.07.010","article-title":"Nullity of a graph in terms of the dimension of cycle space and the number of pendant vertices","volume":"215","author":"Ma","year":"2016","journal-title":"Discrete Appl. Math."},{"key":"10.1016\/j.disc.2026.115119_br0230","doi-asserted-by":"crossref","first-page":"242","DOI":"10.1016\/j.laa.2016.01.036","article-title":"Skew-rank of an oriented graph in terms of matching number","volume":"495","author":"Ma","year":"2016","journal-title":"Linear Algebra Appl."},{"key":"10.1016\/j.disc.2026.115119_br0240","doi-asserted-by":"crossref","first-page":"324","DOI":"10.1016\/j.laa.2015.10.018","article-title":"Hermitian adjacency spectrum and switching equivalence of mixed graphs","volume":"489","author":"Mohar","year":"2016","journal-title":"Linear Algebra Appl."},{"key":"10.1016\/j.disc.2026.115119_br0250","series-title":"Fractional Graph Theory: A Relational Approach to the Theory of Graphs","author":"Scheinerman","year":"1997"},{"key":"10.1016\/j.disc.2026.115119_br0260","doi-asserted-by":"crossref","first-page":"363","DOI":"10.1016\/j.laa.2014.09.034","article-title":"A characterization of graphs G with nullity |V(G)|\u22122m(G)+2\u03bc(G)","volume":"465","author":"Song","year":"2015","journal-title":"Linear Algebra Appl."},{"key":"10.1016\/j.disc.2026.115119_br0270","doi-asserted-by":"crossref","first-page":"276","DOI":"10.1016\/j.dam.2013.09.012","article-title":"Bounds for the matching number, the edge chromatic number and the independence number of a graph in terms of rank","volume":"166","author":"Wang","year":"2014","journal-title":"Discrete Appl. Math."},{"key":"10.1016\/j.disc.2026.115119_br0280","doi-asserted-by":"crossref","first-page":"22","DOI":"10.1016\/j.laa.2017.02.037","article-title":"Mixed graphs with H-rank 3","volume":"524","author":"Wang","year":"2017","journal-title":"Linear Algebra Appl."},{"key":"10.1016\/j.disc.2026.115119_br0290","doi-asserted-by":"crossref","first-page":"2469","DOI":"10.1080\/03081087.2019.1679702","article-title":"Bounds on the nullity, the H-rank and the Hermitian energy of a mixed graph","volume":"69","author":"Wei","year":"2021","journal-title":"Linear Multilinear Algebra"},{"key":"10.1016\/j.disc.2026.115119_br0300","doi-asserted-by":"crossref","first-page":"678","DOI":"10.1007\/s10878-021-00704-6","article-title":"Some mixed graphs with H-rank 4, 6 or 8","volume":"41","author":"Yang","year":"2021","journal-title":"J. Comb. Optim."},{"key":"10.1016\/j.disc.2026.115119_br0310","doi-asserted-by":"crossref","first-page":"314","DOI":"10.1016\/j.laa.2018.06.025","article-title":"An upper bound of the nullity of a graph in terms of order and maximum degree","volume":"555","author":"Zhou","year":"2018","journal-title":"Linear Algebra Appl."},{"key":"10.1016\/j.disc.2026.115119_br0320","doi-asserted-by":"crossref","first-page":"56","DOI":"10.1016\/j.laa.2021.07.005","article-title":"On connected graphs of order n with girth g and nullity n\u2212g","volume":"630","author":"Zhou","year":"2021","journal-title":"Linear Algebra Appl."},{"key":"10.1016\/j.disc.2026.115119_br0330","doi-asserted-by":"crossref","first-page":"93","DOI":"10.1016\/j.dam.2022.01.020","article-title":"Relation between the nullity of a graph and its matching number","volume":"313","author":"Zhou","year":"2022","journal-title":"Discrete Appl. Math."}],"container-title":["Discrete Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/api.elsevier.com\/content\/article\/PII:S0012365X26001433?httpAccept=text\/xml","content-type":"text\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/api.elsevier.com\/content\/article\/PII:S0012365X26001433?httpAccept=text\/plain","content-type":"text\/plain","content-version":"vor","intended-application":"text-mining"}],"deposited":{"date-parts":[[2026,3,19]],"date-time":"2026-03-19T09:45:01Z","timestamp":1773913501000},"score":1,"resource":{"primary":{"URL":"https:\/\/linkinghub.elsevier.com\/retrieve\/pii\/S0012365X26001433"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,8]]},"references-count":32,"journal-issue":{"issue":"8","published-print":{"date-parts":[[2026,8]]}},"alternative-id":["S0012365X26001433"],"URL":"https:\/\/doi.org\/10.1016\/j.disc.2026.115119","relation":{},"ISSN":["0012-365X"],"issn-type":[{"value":"0012-365X","type":"print"}],"subject":[],"published":{"date-parts":[[2026,8]]},"assertion":[{"value":"Elsevier","name":"publisher","label":"This article is maintained by"},{"value":"Improved bounds on the H-rank of a mixed graph in terms of the matching number and fractional matching number","name":"articletitle","label":"Article Title"},{"value":"Discrete Mathematics","name":"journaltitle","label":"Journal Title"},{"value":"https:\/\/doi.org\/10.1016\/j.disc.2026.115119","name":"articlelink","label":"CrossRef DOI link to publisher maintained version"},{"value":"article","name":"content_type","label":"Content Type"},{"value":"\u00a9 2026 Elsevier B.V. All rights are reserved, including those for text and data mining, AI training, and similar technologies.","name":"copyright","label":"Copyright"}],"article-number":"115119"}}