{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,18]],"date-time":"2026-05-18T00:02:53Z","timestamp":1779062573243,"version":"3.51.4"},"reference-count":26,"publisher":"Springer Science and Business Media LLC","issue":"6","license":[{"start":{"date-parts":[[2021,8,13]],"date-time":"2021-08-13T00:00:00Z","timestamp":1628812800000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springer.com\/tdm"},{"start":{"date-parts":[[2021,8,13]],"date-time":"2021-08-13T00:00:00Z","timestamp":1628812800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springer.com\/tdm"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Comp. Appl. Math."],"published-print":{"date-parts":[[2021,9]]},"DOI":"10.1007\/s40314-021-01589-1","type":"journal-article","created":{"date-parts":[[2021,8,13]],"date-time":"2021-08-13T22:02:32Z","timestamp":1628892152000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["On the Randi\u0107 incidence energy of graphs"],"prefix":"10.1007","volume":"40","author":[{"given":"M.","family":"Mateji\u0107","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"\u015e. B.","family":"Bozkurt Altinda\u011f","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"E.","family":"Milovanovi\u0107","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2209-9606","authenticated-orcid":false,"given":"I.","family":"Milovanovi\u0107","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2021,8,13]]},"reference":[{"issue":"2","key":"1589_CR1","first-page":"171","volume":"4","author":"\u015eB Bozkurt Alt\u0131nda\u011f","year":"2019","unstructured":"Bozkurt Alt\u0131nda\u011f \u015eB (2019) Note on the sum of powers of normalized signless Laplacian eigenvalues of graphs. Math Interdisc Res 4(2):171\u2013182","journal-title":"Math Interdisc Res"},{"key":"1589_CR2","first-page":"135","volume":"11","author":"\u015eB Bozkurt Alt\u0131nda\u011f","year":"2021","unstructured":"Bozkurt Alt\u0131nda\u011f \u015eB (2021) Sum of powers of normalized signless Laplacian eigenvalues and Randi\u0107 (normalized) incidence energy of graphs. Bull Int Math Virtual Inst 11:135\u2013146","journal-title":"Bull Int Math Virtual Inst"},{"issue":"1","key":"1589_CR3","first-page":"239","volume":"64","author":"\u015eB Bozkurt","year":"2010","unstructured":"Bozkurt \u015eB, Gungor AD, Gutman I, Cevik AS (2010) Randi\u0107 matrix and Randi\u0107 energy. MATCH Commun Math Comput Chem 64(1):239\u2013250","journal-title":"MATCH Commun Math Comput Chem"},{"key":"1589_CR4","doi-asserted-by":"publisher","first-page":"172","DOI":"10.1016\/j.laa.2010.02.002","volume":"433","author":"M Cavers","year":"2010","unstructured":"Cavers M, Fallat S, Kirkland S (2010) On the normalized Laplacian energy and general Randi\u0107 index $$R_{-1}$$ of graphs. Linear Algebra Appl 433:172\u2013190","journal-title":"Linear Algebra Appl"},{"issue":"11","key":"1589_CR5","doi-asserted-by":"publisher","first-page":"4510","DOI":"10.1016\/j.laa.2013.01.003","volume":"438","author":"B Cheng","year":"2013","unstructured":"Cheng B, Liu B (2013) The normalized incidence energy of a graph. Linear Algebra Appl 438(11):4510\u20134519","journal-title":"Linear Algebra Appl"},{"key":"1589_CR6","volume-title":"Spectral graph theory","author":"FRK Chung","year":"1997","unstructured":"Chung FRK (1997) Spectral graph theory. American Mathematical Society, Providence"},{"key":"1589_CR7","volume-title":"Spectra of graphs: theory and applications","author":"D Cvetkovi\u0107","year":"1980","unstructured":"Cvetkovi\u0107 D, Doob M, Sachs H (1980) Spectra of graphs: theory and applications. Academic press, New York"},{"issue":"1","key":"1589_CR8","doi-asserted-by":"publisher","first-page":"155","DOI":"10.1016\/j.laa.2007.01.009","volume":"423","author":"D Cvetkovi\u0107","year":"2007","unstructured":"Cvetkovi\u0107 D, Rowlinson P, Simi\u0107 S (2007) Signless Laplacians of finite graphs. Linear Algebra Appl 423(1):155\u2013171","journal-title":"Linear Algebra Appl"},{"key":"1589_CR9","first-page":"663","volume":"70","author":"KC Das","year":"2013","unstructured":"Das KC, Mojallal SA, Gutman I (2013) Improving McClelland\u2019s lower bound for energy. MATCH Commun Math Comput Chem 70:663\u2013668","journal-title":"MATCH Commun Math Comput Chem"},{"issue":"4","key":"1589_CR10","first-page":"1","volume":"3","author":"R Gu","year":"2014","unstructured":"Gu R, Huang F, Li X (2014) Randi\u0107 incidence energy of graphs. Trans Comb 3(4):1\u20139","journal-title":"Trans Comb"},{"key":"1589_CR11","first-page":"1","volume":"103","author":"I Gutman","year":"1978","unstructured":"Gutman I (1978) The energy of a graph. Ber Math Stat Sekt Forsch Graz 103:1\u201322","journal-title":"Ber Math Stat Sekt Forsch Graz"},{"key":"1589_CR12","doi-asserted-by":"publisher","first-page":"535","DOI":"10.1016\/0009-2614(72)85099-1","volume":"17","author":"I Gutman","year":"1972","unstructured":"Gutman I, Trinajsti\u0107 N (1972) Graph theory and molecular orbitals. Total $$\\pi $$-electron energy of alternant hydrocarbons. Chem Phys Lett 17:535\u2013538","journal-title":"Chem Phys Lett"},{"key":"1589_CR13","first-page":"573","volume":"62","author":"I Gutman","year":"2009","unstructured":"Gutman I, Kiani D, Mirzakhah M (2009) On incidence energy of graphs. MATCH Commun Math Comput Chem 62:573\u2013580","journal-title":"MATCH Commun Math Comput Chem"},{"key":"1589_CR14","doi-asserted-by":"publisher","first-page":"175","DOI":"10.1007\/BF02418571","volume":"30","author":"JLW Jensen","year":"1906","unstructured":"Jensen JLW (1906) Sur les functions convexes of les integralities entre les values moyennes. Acta Math 30:175\u2013193","journal-title":"Acta Math"},{"key":"1589_CR15","first-page":"1","volume":"316","author":"J Li","year":"2014","unstructured":"Li J, Guo JM, Shiu WC (2014) Bounds on normalized Laplacian eigenvalues of graphs. J Inequal Appl 316:1\u20138","journal-title":"J Inequal Appl"},{"issue":"2","key":"1589_CR16","first-page":"355","volume":"59","author":"J Liu","year":"2008","unstructured":"Liu J, Liu B (2008) A Laplacian-energy-like invariant of a graph. MATCH Commun Math Comput Chem 59(2):355\u2013372","journal-title":"MATCH Commun Math Comput Chem"},{"issue":"3","key":"1589_CR17","first-page":"913","volume":"68","author":"B Liu","year":"2012","unstructured":"Liu B, Huang Y, Feng J (2012) A note on the Randi\u0107 spectral radius. MATCH Commun Math Comput Chem 68(3):913\u2013916","journal-title":"MATCH Commun Math Comput Chem"},{"key":"1589_CR18","doi-asserted-by":"publisher","first-page":"143","DOI":"10.1016\/0024-3795(94)90486-3","volume":"197\u2013198","author":"R Merris","year":"1994","unstructured":"Merris R (1994) Laplacian matrices of graphs, a survey. Linear Algebra Appl 197\u2013198:143\u2013176","journal-title":"Linear Algebra Appl"},{"key":"1589_CR19","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-99970-3","volume-title":"Analytic inequalities","author":"DS Mitrinovi\u0107","year":"1970","unstructured":"Mitrinovi\u0107 DS, Vasi\u0107 PM (1970) Analytic inequalities. Springer, Berlin"},{"key":"1589_CR20","doi-asserted-by":"publisher","first-page":"1472","DOI":"10.1016\/j.jmaa.2006.03.072","volume":"326","author":"V Nikiforov","year":"2007","unstructured":"Nikiforov V (2007) The energy of graphs and matrices. J Math Anal Appl 326:1472\u20131475","journal-title":"J Math Anal Appl"},{"issue":"2","key":"1589_CR21","doi-asserted-by":"publisher","first-page":"1249","DOI":"10.1016\/j.jmaa.2015.10.055","volume":"435","author":"L Qiao","year":"2016","unstructured":"Qiao L, Zhang S, Ning B, Li J (2016) Coulson-type integral formulas for the general Laplacian energy-like invariant of graphs. J Math Anal Appl 435(2):1249\u20131261","journal-title":"J Math Anal Appl"},{"key":"1589_CR22","doi-asserted-by":"publisher","first-page":"6609","DOI":"10.1021\/ja00856a001","volume":"97","author":"M Randi\u0107","year":"1975","unstructured":"Randi\u0107 M (1975) On characterization of molecular branching. J Am Chem Soc 97:6609\u20136615","journal-title":"J Am Chem Soc"},{"issue":"16","key":"1589_CR23","doi-asserted-by":"publisher","first-page":"5238","DOI":"10.1016\/j.disc.2009.03.036","volume":"309","author":"L Shi","year":"2009","unstructured":"Shi L (2009) Bounds on Randi\u0107 indices. Discrete Math 309(16):5238\u20135241","journal-title":"Discrete Math"},{"issue":"12","key":"1589_CR24","doi-asserted-by":"publisher","first-page":"4056","DOI":"10.1016\/j.laa.2013.10.028","volume":"439","author":"L Shi","year":"2013","unstructured":"Shi L, Wang H (2013) The Laplacian incidence energy of graphs. Linear Algebra Appl 439(12):4056\u20134062","journal-title":"Linear Algebra Appl"},{"key":"1589_CR25","unstructured":"Woods C (2020) My Favorite application using graph eigenvalues: graph energy. http:\/\/www.math.ucsd.edu\/~fan\/teach\/262\/13\/262notes\/Woods_Midterm.pdf. Accessed 20 Sep 2020"},{"key":"1589_CR26","unstructured":"Zumstein P (2005) Comparison of spectral methods through the adjacency matrix and the Laplacian of a graph, Th. Diploma, ETH Z\u00fcrich"}],"container-title":["Computational and Applied Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s40314-021-01589-1.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s40314-021-01589-1\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s40314-021-01589-1.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,9,24]],"date-time":"2021-09-24T17:30:43Z","timestamp":1632504643000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s40314-021-01589-1"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,8,13]]},"references-count":26,"journal-issue":{"issue":"6","published-print":{"date-parts":[[2021,9]]}},"alternative-id":["1589"],"URL":"https:\/\/doi.org\/10.1007\/s40314-021-01589-1","relation":{},"ISSN":["2238-3603","1807-0302"],"issn-type":[{"value":"2238-3603","type":"print"},{"value":"1807-0302","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,8,13]]},"assertion":[{"value":"24 September 2020","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"21 July 2021","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"22 July 2021","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"13 August 2021","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}],"article-number":"209"}}