{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,11]],"date-time":"2026-02-11T13:16:14Z","timestamp":1770815774391,"version":"3.50.1"},"reference-count":21,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2022,12,27]],"date-time":"2022-12-27T00:00:00Z","timestamp":1672099200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"the Qinghai Key Laboratory of Internet of Things Project","award":["2017-ZJ-Y21"],"award-info":[{"award-number":["2017-ZJ-Y21"]}]},{"name":"the Qinghai Key Laboratory of Internet of Things Project","award":["2021R1F1A1050646"],"award-info":[{"award-number":["2021R1F1A1050646"]}]},{"name":"National Research Foundation funded by the Korean government","award":["2017-ZJ-Y21"],"award-info":[{"award-number":["2017-ZJ-Y21"]}]},{"name":"National Research Foundation funded by the Korean government","award":["2021R1F1A1050646"],"award-info":[{"award-number":["2021R1F1A1050646"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>The vertex-degree function index\u00a0Hf(\u0393) is defined as Hf(\u0393)=\u2211v\u2208V(\u0393)f(d(v)) for a function f(x) defined on non-negative real numbers. In this paper, we determine the extremal graphs with the maximum (minimum) vertex degree function index in the set of all n-vertex chemical trees, trees, and connected graphs. We also present the Nordhaus\u2013Gaddum-type results for Hf(\u0393)+Hf(\u0393\u00af) and Hf(\u0393)\u00b7Hf(\u0393\u00af).<\/jats:p>","DOI":"10.3390\/axioms12010031","type":"journal-article","created":{"date-parts":[[2022,12,28]],"date-time":"2022-12-28T05:38:43Z","timestamp":1672205923000},"page":"31","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":6,"title":["Extremal Graphs to Vertex Degree Function Index for Convex Functions"],"prefix":"10.3390","volume":"12","author":[{"given":"Dong","family":"He","sequence":"first","affiliation":[{"name":"School of Mathematics and Statistis, Qinghai Normal University, Xining 810008, China"}]},{"given":"Zhen","family":"Ji","sequence":"additional","affiliation":[{"name":"School of Computer, Qinghai Normal University, Xining 810008, China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2523-7090","authenticated-orcid":false,"given":"Chenxu","family":"Yang","sequence":"additional","affiliation":[{"name":"School of Computer, Qinghai Normal University, Xining 810008, China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2576-160X","authenticated-orcid":false,"given":"Kinkar Chandra","family":"Das","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Sungkyunkwan University, Suwon 16419, Republic of Korea"}]}],"member":"1968","published-online":{"date-parts":[[2022,12,27]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"218","DOI":"10.1016\/j.dam.2019.06.005","article-title":"Unified extremal results of topological indices and spectral invariants of graphs","volume":"271","author":"Yao","year":"2019","journal-title":"Discrete Appl. Math."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"535","DOI":"10.1016\/0009-2614(72)85099-1","article-title":"Graph theory and molecular orbitals, Total \u03c6-electron energy of alternant hydrocarbons","volume":"17","author":"Gutman","year":"1972","journal-title":"Chem. Phys. Lett."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"1184","DOI":"10.1007\/s10910-015-0480-z","article-title":"A forgotten topological index","volume":"53","author":"Furtula","year":"2015","journal-title":"J. Math. Chem."},{"key":"ref_4","first-page":"425","article-title":"On molecular graphs with smallest and greatest zeroth-order general Randi\u0107 index","volume":"54","author":"Hu","year":"2005","journal-title":"MATCH Commun. Math. Comput. Chem."},{"key":"ref_5","first-page":"195","article-title":"A unified approach to the extremal trees for different indices","volume":"54","author":"Li","year":"2005","journal-title":"MATCH Commun. Math. Comput. Chem."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"41","DOI":"10.18514\/MMN.2022.2382","article-title":"On the extremal graphs for second Zagreb index with fixed number of vertices and cyclomatic number","volume":"23","author":"Ali","year":"2022","journal-title":"Miskolc Math. Notes"},{"key":"ref_7","first-page":"141","article-title":"First Zagreb index, k-connectivity, \u03b2-deficiency and k-hamiltonicity of graphs","volume":"80","author":"An","year":"2018","journal-title":"MATCH Commun. Math. Comput. Chem."},{"key":"ref_8","first-page":"585","article-title":"Comparison between the zeroth-order Randi\u0107 index and the sum-connectivity index","volume":"274","author":"Das","year":"2016","journal-title":"Appl. Math. Comput."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"567","DOI":"10.1007\/s11464-015-0431-9","article-title":"On Zagreb indices of graphs","volume":"10","author":"Das","year":"2015","journal-title":"Front. Math. China"},{"key":"ref_10","first-page":"597","article-title":"A unified approach to the extremal Zagreb indices for trees, unicyclic graphs and bicyclic graphs","volume":"57","author":"Deng","year":"2007","journal-title":"MATCH Commun. Math. Comput. Chem"},{"key":"ref_11","first-page":"295","article-title":"On Zagreb indices of graphs","volume":"85","author":"Horoldagva","year":"2021","journal-title":"MATCH Commun. Math. Comput. Chem."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"19","DOI":"10.1016\/j.dam.2018.10.009","article-title":"On the extremal graphs for general sum-connectivity index \u03c7\u03b1 with given cyclomatic number when \u03b1>1","volume":"257","author":"Ali","year":"2019","journal-title":"Discrete Appl. Math."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"1044","DOI":"10.1016\/j.dam.2006.11.008","article-title":"Connected (n,m)-graphs with minimum and maximum zeroth-order general Randi\u0107 index","volume":"155","author":"Hu","year":"2007","journal-title":"Discrete Appl. Math."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"57","DOI":"10.1016\/j.disc.2004.04.007","article-title":"Maximizing the sum of the squares of the degrees of a graph","volume":"285","author":"Das","year":"2004","journal-title":"Discrete Math."},{"key":"ref_15","first-page":"163","article-title":"(n,m)-graphs with maximum zeroth-order general Randi\u0107 index for alpha is an element of \u03b1\u2208(\u22121, 0)","volume":"62","author":"Li","year":"2009","journal-title":"MATCH Commun. Math. Comput. Chem."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"2938","DOI":"10.1016\/j.dam.2009.02.014","article-title":"More on connected (n,m)-graphs with minimum and maximum zeroth-order general Randi\u0107 index","volume":"157","year":"2009","journal-title":"Discrete Appl. Math."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"109","DOI":"10.46793\/match.87-1.109T","article-title":"Graphs with given cyclomatic number extremal relatively to vertex degree function index for convex functions","volume":"87","author":"Tomescu","year":"2022","journal-title":"MATCH Commun. Math. Comput. Chem."},{"key":"ref_18","first-page":"285","article-title":"Properties of connected (n,m)-graphs extremal relatively to vertex degree function index for convex functions","volume":"85","author":"Tomescu","year":"2021","journal-title":"MATCH Commun. Math. Comput. Chem."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"466","DOI":"10.1016\/j.dam.2011.12.018","article-title":"A survey of Nordhaus\u2013Gaddum type relations","volume":"161","author":"Aouchiche","year":"2013","journal-title":"Discrete Appl. Math."},{"key":"ref_20","unstructured":"Gutman, I., Furtula, B., Das, K.C., Milovanovi\u0107, E., and Milovanovixcx, I. (2017). Nordhaus\u2013Gaddum Type Results in Chemical Graph Theory, University Kragujevac. Bounds in Chemical Graph Theory\u2014Advances."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"175","DOI":"10.1007\/BF02418571","article-title":"Sur les fonctions convexes et les in\u00e9galit\u00e9s entre les valeurs moyennes","volume":"30","author":"Jensen","year":"1906","journal-title":"Acta Math."}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/12\/1\/31\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T01:52:44Z","timestamp":1760147564000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/12\/1\/31"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,12,27]]},"references-count":21,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2023,1]]}},"alternative-id":["axioms12010031"],"URL":"https:\/\/doi.org\/10.3390\/axioms12010031","relation":{},"ISSN":["2075-1680"],"issn-type":[{"value":"2075-1680","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,12,27]]}}}