{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,3]],"date-time":"2026-03-03T05:57:13Z","timestamp":1772517433548,"version":"3.50.1"},"reference-count":25,"publisher":"Springer Science and Business Media LLC","issue":"7","license":[{"start":{"date-parts":[[2026,3,3]],"date-time":"2026-03-03T00:00:00Z","timestamp":1772496000000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2026,3,3]],"date-time":"2026-03-03T00:00:00Z","timestamp":1772496000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["12071194, 11571155"],"award-info":[{"award-number":["12071194, 11571155"]}],"id":[{"id":"10.13039\/501100001809","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":[[2026,8]]},"DOI":"10.1007\/s40314-026-03665-w","type":"journal-article","created":{"date-parts":[[2026,3,3]],"date-time":"2026-03-03T05:12:58Z","timestamp":1772514778000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["On graphs with second largest closeness eigenvalue at most zero"],"prefix":"10.1007","volume":"45","author":[{"given":"Qaisar","family":"Farhad","sequence":"first","affiliation":[]},{"given":"Fazal","family":"Hayat","sequence":"additional","affiliation":[]},{"given":"Shou-Jun","family":"Xu","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2026,3,3]]},"reference":[{"issue":"5","key":"3665_CR1","doi-asserted-by":"publisher","first-page":"447","DOI":"10.1080\/00029890.1998.12004907","volume":"105","author":"B Anderson","year":"1998","unstructured":"Anderson B, Jackson J, Sitharam M (1998) Descartes\u2019 rule of signs revisited. Am Math Mon 105(5):447\u2013451. https:\/\/doi.org\/10.1080\/00029890.1998.12004907","journal-title":"Am Math Mon"},{"key":"3665_CR2","doi-asserted-by":"publisher","first-page":"46","DOI":"10.1016\/j.dam.2021.09.019","volume":"306","author":"M Aouchiche","year":"2022","unstructured":"Aouchiche M, ElHallaoui I (2022) Minimum values of the second largest Q-eigenvalue. Discrete Appl Math 306:46\u201351. https:\/\/doi.org\/10.1016\/j.dam.2021.09.019","journal-title":"Discrete Appl Math"},{"key":"3665_CR3","doi-asserted-by":"publisher","first-page":"799","DOI":"10.1016\/j.laa.2005.12.023","volume":"416","author":"RB Bapat","year":"2006","unstructured":"Bapat RB, Lal AK, Pati S (2006) A q-analogue of the distance matrix of a tree. Linear Algebra Appl 416:799\u2013814. https:\/\/doi.org\/10.1016\/j.laa.2005.12.023","journal-title":"Linear Algebra Appl"},{"key":"3665_CR4","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4614-1939-6","volume-title":"Spectra of graphs","author":"AE Brouwer","year":"2012","unstructured":"Brouwer AE, Haemers WH (2012) Spectra of graphs. Springer, New York. https:\/\/doi.org\/10.1007\/978-1-4614-1939-6"},{"key":"3665_CR5","volume-title":"Distance in graphs","author":"F Buckley","year":"1990","unstructured":"Buckley F, Harary F (1990) Distance in graphs. Addison-Wesley, Redwood City"},{"key":"3665_CR6","doi-asserted-by":"publisher","first-page":"839","DOI":"10.1007\/s40305-022-00405-9","volume":"11","author":"M Cheng","year":"2022","unstructured":"Cheng M, Zhou B (2022) Residual closeness of graphs with given parameters. J Oper Res Soc China 11:839\u2013856. https:\/\/doi.org\/10.1007\/s40305-022-00405-9","journal-title":"J Oper Res Soc China"},{"key":"3665_CR7","doi-asserted-by":"publisher","first-page":"343","DOI":"10.1016\/j.laa.2019.06.027","volume":"580","author":"Y Chen","year":"2019","unstructured":"Chen Y, Li D, Meng J (2019) On the second largest $$\\text{ A}_\\alpha $$-eigenvalues of graphs. Linear Algebra Appl 580:343\u2013358. https:\/\/doi.org\/10.1016\/j.laa.2019.06.027","journal-title":"Linear Algebra Appl"},{"key":"3665_CR8","doi-asserted-by":"publisher","unstructured":"Cvetkovic D, Rowlinson P, Simic S (2010) An introduction to the theory of graph spectra. https:\/\/doi.org\/10.1017\/CBO9780511801518","DOI":"10.1017\/CBO9780511801518"},{"key":"3665_CR9","doi-asserted-by":"publisher","first-page":"556","DOI":"10.1016\/j.physa.2005.12.020","volume":"365","author":"C Dangalchev","year":"2006","unstructured":"Dangalchev C (2006) Residual closeness in networks. Physica A 365:556\u2013564. https:\/\/doi.org\/10.1016\/j.physa.2005.12.020","journal-title":"Physica A"},{"issue":"1","key":"3665_CR10","doi-asserted-by":"publisher","first-page":"1","DOI":"10.3233\/FI-2020-1960","volume":"176","author":"C Dangalchev","year":"2020","unstructured":"Dangalchev C (2020) Additional closeness and networks growth. Fundam Inform 176(1):1\u201315. https:\/\/doi.org\/10.3233\/FI-2020-1960","journal-title":"Fundam Inform"},{"issue":"1","key":"3665_CR11","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1142\/S0129054122500149","volume":"33","author":"C Dangalchev","year":"2022","unstructured":"Dangalchev C (2022) Additional closeness of cycle graphs. Int J Found Comput Sci 33(1):1\u201321. https:\/\/doi.org\/10.1142\/S0129054122500149","journal-title":"Int J Found Comput Sci"},{"issue":"3","key":"3665_CR12","doi-asserted-by":"publisher","first-page":"215","DOI":"10.1016\/0378-8733(78)90021-7","volume":"1","author":"L Freeman","year":"1978","unstructured":"Freeman L (1978) Centrality in social networks: conceptual clarification. Soc Netw 1(3):215\u2013239. https:\/\/doi.org\/10.1016\/0378-8733(78)90021-7","journal-title":"Soc Netw"},{"key":"3665_CR13","doi-asserted-by":"publisher","DOI":"10.1016\/j.disc.2024.114082","volume":"347","author":"H Guo","year":"2024","unstructured":"Guo H, Zhou B (2024) Graphs for which the second largest distance eigenvalue is less than 1\/2. Discrete Math 347:114082. https:\/\/doi.org\/10.1016\/j.disc.2024.114082","journal-title":"Discrete Math"},{"key":"3665_CR14","doi-asserted-by":"publisher","first-page":"810","DOI":"10.3390\/axioms13120810","volume":"13","author":"F Hayat","year":"2024","unstructured":"Hayat F, Otera DE (2024a) Extremal k-connected graphs with maximum closeness. Axioms 13:810. https:\/\/doi.org\/10.3390\/axioms13120810","journal-title":"Axioms"},{"key":"3665_CR15","doi-asserted-by":"publisher","first-page":"20","DOI":"10.3390\/math12132039","volume":"12","author":"F Hayat","year":"2024","unstructured":"Hayat F, Otera DE (2024b) Maximizing closeness in bipartite networks: a graph-theoretic analysis. Mathematics 12:20\u201339. https:\/\/doi.org\/10.3390\/math12132039","journal-title":"Mathematics"},{"issue":"19","key":"3665_CR16","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.87.198701","volume":"87","author":"V Latora","year":"2001","unstructured":"Latora V, Marchiori M (2001) Efficient behavior of small-world networks. Phys Rev Lett 87(19):198701. https:\/\/doi.org\/10.1103\/PhysRevLett.87.198701","journal-title":"Phys Rev Lett"},{"key":"3665_CR17","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1007\/s10878-023-01042-5","volume":"45","author":"C Li","year":"2023","unstructured":"Li C, Xu L, Zhou B (2023) On the residual closeness of graphs with cut vertices. J Comb Optim 45:1\u201315. https:\/\/doi.org\/10.1007\/s10878-023-01042-5","journal-title":"J Comb Optim"},{"key":"3665_CR18","doi-asserted-by":"publisher","first-page":"989","DOI":"10.1007\/s10801-021-01080-4","volume":"55","author":"J Siemons","year":"2022","unstructured":"Siemons J, Zalesski A (2022) On the second largest eigenvalue of some Cayley graphs of the symmetric group. J Algebr Comb 55:989\u20131005","journal-title":"J Algebr Comb"},{"key":"3665_CR19","doi-asserted-by":"publisher","first-page":"118","DOI":"10.1016\/j.dam.2019.10.010","volume":"279","author":"O Suil","year":"2020","unstructured":"Suil O (2020) The second largest eigenvalue and vertex-connectivity of regular multigraphs. Discrete Appl Math 279:118\u2013124. https:\/\/doi.org\/10.1016\/j.dam.2019.10.010","journal-title":"Discrete Appl Math"},{"issue":"5","key":"3665_CR20","doi-asserted-by":"publisher","first-page":"1156","DOI":"10.1093\/comjnl\/bxac004","volume":"66","author":"Y Wang","year":"2023","unstructured":"Wang Y, Zhou B (2023) Residual closeness, matching number, and chromatic number. Comput J 66(5):1156\u20131166. https:\/\/doi.org\/10.1093\/comjnl\/bxac004","journal-title":"Comput J"},{"issue":"3","key":"3665_CR21","doi-asserted-by":"publisher","first-page":"311","DOI":"10.1016\/j.aam.2006.04.002","volume":"39","author":"W Yan","year":"2007","unstructured":"Yan W, Yeh Y-N (2007) The determinants of q-distance matrices of trees and two quantities relating to permutations. Adv Appl Math 39(3):311\u2013321. https:\/\/doi.org\/10.1016\/j.aam.2006.04.002","journal-title":"Adv Appl Math"},{"issue":"4","key":"3665_CR22","doi-asserted-by":"publisher","first-page":"3525","DOI":"10.1051\/ro\/2022161","volume":"56","author":"L Zheng","year":"2022","unstructured":"Zheng L, Zhou B (2022a) Spectra of closeness Laplacian and closeness signless Laplacian of graphs. RAIRO Oper Res 56(4):3525\u20133543. https:\/\/doi.org\/10.1051\/ro\/2022161","journal-title":"RAIRO Oper Res"},{"issue":"4","key":"3665_CR23","doi-asserted-by":"publisher","first-page":"2651","DOI":"10.1051\/ro\/2022125","volume":"56","author":"L Zheng","year":"2022","unstructured":"Zheng L, Zhou B (2022b) On the spectral closeness and residual spectral closeness of graphs. RAIRO Oper Res 56(4):2651\u20132668. https:\/\/doi.org\/10.1051\/ro\/2022125","journal-title":"RAIRO Oper Res"},{"issue":"3","key":"3665_CR24","doi-asserted-by":"publisher","first-page":"741","DOI":"10.1007\/s10801-023-01270-2","volume":"58","author":"L Zheng","year":"2023","unstructured":"Zheng L, Zhou B (2023) The closeness eigenvalues of graphs. J Algebr Comb 58(3):741\u2013760. https:\/\/doi.org\/10.1007\/s10801-023-01270-2","journal-title":"J Algebr Comb"},{"issue":"10","key":"3665_CR25","doi-asserted-by":"publisher","first-page":"1591","DOI":"10.1016\/j.disc.2012.02.002","volume":"312","author":"J Zhou","year":"2012","unstructured":"Zhou J, Bu C (2012) Laplacian spectral characterization of some graphs obtained by product operation. Discrete Math 312(10):1591\u20131595. https:\/\/doi.org\/10.1016\/j.disc.2012.02.002","journal-title":"Discrete Math"}],"container-title":["Computational and Applied Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s40314-026-03665-w.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s40314-026-03665-w","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s40314-026-03665-w.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,3,3]],"date-time":"2026-03-03T05:12:59Z","timestamp":1772514779000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s40314-026-03665-w"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,3,3]]},"references-count":25,"journal-issue":{"issue":"7","published-print":{"date-parts":[[2026,8]]}},"alternative-id":["3665"],"URL":"https:\/\/doi.org\/10.1007\/s40314-026-03665-w","relation":{},"ISSN":["2238-3603","1807-0302"],"issn-type":[{"value":"2238-3603","type":"print"},{"value":"1807-0302","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,3,3]]},"assertion":[{"value":"30 April 2025","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"7 January 2026","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"9 February 2026","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"3 March 2026","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 conflict of interest.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}}],"article-number":"271"}}