{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T14:38:32Z","timestamp":1753886312217,"version":"3.41.2"},"reference-count":25,"publisher":"Wiley","issue":"1","license":[{"start":{"date-parts":[[2012,4,23]],"date-time":"2012-04-23T00:00:00Z","timestamp":1335139200000},"content-version":"vor","delay-in-days":113,"URL":"http:\/\/creativecommons.org\/licenses\/by\/3.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["11126255"],"award-info":[{"award-number":["11126255"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["onlinelibrary.wiley.com"],"crossmark-restriction":true},"short-container-title":["Journal of Applied Mathematics"],"published-print":{"date-parts":[[2012,1]]},"abstract":"<jats:p>A (<jats:italic>m<\/jats:italic>, <jats:italic>n<\/jats:italic>)\u2010bipartite graph is a bipartite graph such that one bipartition has <jats:italic>m<\/jats:italic> vertices and the other bipartition has <jats:italic>n<\/jats:italic> vertices. The tree dumbbell <jats:italic>D<\/jats:italic>(<jats:italic>n<\/jats:italic>, <jats:italic>a<\/jats:italic>, <jats:italic>b<\/jats:italic>) consists of the path <jats:italic>P<\/jats:italic><jats:sub><jats:italic>n<\/jats:italic>\u2212<jats:italic>a<\/jats:italic>\u2212<jats:italic>b<\/jats:italic><\/jats:sub> together with <jats:italic>a<\/jats:italic> independent vertices adjacent to one pendent vertex of <jats:italic>P<\/jats:italic><jats:sub><jats:italic>n<\/jats:italic>\u2212<jats:italic>a<\/jats:italic>\u2212<jats:italic>b<\/jats:italic><\/jats:sub> and <jats:italic>b<\/jats:italic> independent vertices adjacent to the other pendent vertex of <jats:italic>P<\/jats:italic><jats:sub><jats:italic>n<\/jats:italic>\u2212<jats:italic>a<\/jats:italic>\u2212<jats:italic>b<\/jats:italic><\/jats:sub>. In this paper, firstly, we show that, among (<jats:italic>m<\/jats:italic>, <jats:italic>n<\/jats:italic>)\u2010bipartite graphs (<jats:italic>m<\/jats:italic> \u2264 <jats:italic>n<\/jats:italic>), the complete bipartite graph <jats:italic>K<\/jats:italic><jats:sub><jats:italic>m<\/jats:italic>,<jats:italic>n<\/jats:italic><\/jats:sub> has minimal Kirchhoff index and the tree dumbbell <jats:italic>D<\/jats:italic>(<jats:italic>m<\/jats:italic> + <jats:italic>n<\/jats:italic>, \u230a<jats:italic>n<\/jats:italic> \u2212 <jats:italic>(m<\/jats:italic> + 1)\/2\u230b, \u2308<jats:italic>n<\/jats:italic> \u2212 <jats:italic>(m<\/jats:italic> + 1)\/2\u2309) has maximal Kirchhoff index. Then, we show that, among all bipartite graphs of order <jats:italic>l<\/jats:italic>, the complete bipartite graph <jats:italic>K<\/jats:italic><jats:sub>\u230a<jats:italic>l<\/jats:italic>\/2\u230b,<jats:italic>l<\/jats:italic>\u2212\u230a<jats:italic>l<\/jats:italic>\/2\u230b<\/jats:sub> has minimal Kirchhoff index and the path <jats:italic>P<\/jats:italic><jats:sub><jats:italic>l<\/jats:italic><\/jats:sub> has maximal Kirchhoff index, respectively. Finally, bonds for the Kirchhoff index of (<jats:italic>m<\/jats:italic>, <jats:italic>n<\/jats:italic>)\u2010bipartite graphs and bipartite graphs of order <jats:italic>l<\/jats:italic>\n are obtained by computing the Kirchhoff index of these extremal graphs.<\/jats:p>","DOI":"10.1155\/2012\/195242","type":"journal-article","created":{"date-parts":[[2012,4,23]],"date-time":"2012-04-23T21:41:00Z","timestamp":1335217260000},"update-policy":"https:\/\/doi.org\/10.1002\/crossmark_policy","source":"Crossref","is-referenced-by-count":6,"title":["Bounds for the Kirchhoff Index of Bipartite Graphs"],"prefix":"10.1155","volume":"2012","author":[{"given":"Yujun","family":"Yang","sequence":"first","affiliation":[]}],"member":"311","published-online":{"date-parts":[[2012,4,23]]},"reference":[{"key":"e_1_2_5_1_2","doi-asserted-by":"publisher","DOI":"10.1021\/ja01193a005"},{"key":"e_1_2_5_2_2","doi-asserted-by":"publisher","DOI":"10.1007\/BF01164627"},{"key":"e_1_2_5_3_2","doi-asserted-by":"publisher","DOI":"10.1002\/qua.560500102"},{"key":"e_1_2_5_4_2","doi-asserted-by":"publisher","DOI":"10.1007\/s00214-003-0460-4"},{"key":"e_1_2_5_5_2","first-page":"683","article-title":"The second maximal and minimal Kirchhoff indices of unicyclic graphs","volume":"61","author":"Zhang W.","year":"2009","journal-title":"Communications in Mathematical and in Computer Chemistry"},{"key":"e_1_2_5_6_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.dam.2009.03.007"},{"key":"e_1_2_5_7_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.laa.2009.05.032"},{"key":"e_1_2_5_8_2","doi-asserted-by":"publisher","DOI":"10.1002\/qua.22318"},{"key":"e_1_2_5_9_2","doi-asserted-by":"crossref","first-page":"865","DOI":"10.1515\/zna-2010-1014","article-title":"Kirchhoff index of cyclopolyacenes","volume":"65","author":"Wang Y.","year":"2010","journal-title":"Zeitschrift fur Naturforschung"},{"key":"e_1_2_5_10_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.dam.2011.06.027"},{"key":"e_1_2_5_11_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.dam.2011.11.011"},{"key":"e_1_2_5_12_2","doi-asserted-by":"publisher","DOI":"10.1007\/s10910-011-9845-0"},{"key":"e_1_2_5_13_2","doi-asserted-by":"publisher","DOI":"10.1080\/03081081003794233"},{"key":"e_1_2_5_14_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.cplett.2008.02.060"},{"key":"e_1_2_5_15_2","doi-asserted-by":"publisher","DOI":"10.1007\/s10910-008-9459-3"},{"key":"e_1_2_5_16_2","doi-asserted-by":"publisher","DOI":"10.1002\/qua.21915"},{"key":"e_1_2_5_17_2","doi-asserted-by":"publisher","DOI":"10.1002\/(SICI)1097-461X(1999)71:3<217::AID-QUA1>3.0.CO;2-C"},{"key":"e_1_2_5_18_2","doi-asserted-by":"publisher","DOI":"10.1002\/1097-461X(2001)81:1<29::AID-QUA6>3.0.CO;2-Y"},{"key":"e_1_2_5_19_2","doi-asserted-by":"publisher","DOI":"10.1002\/qua.21068"},{"key":"e_1_2_5_20_2","first-page":"713","article-title":"The extremal Kirchhoff index of a class of unicyclic graphs","volume":"61","author":"Guo Q.","year":"2009","journal-title":"Communications in Mathematical and in Computer Chemistry"},{"key":"e_1_2_5_21_2","first-page":"697","article-title":"Bicyclic graphs with extremal Kirchhoff index","volume":"61","author":"Zhang H.","year":"2009","journal-title":"Communications in Mathematical and in Computer Chemistry"},{"key":"e_1_2_5_22_2","unstructured":"FengL. 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