{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,27]],"date-time":"2026-05-27T10:24:44Z","timestamp":1779877484776,"version":"3.53.1"},"reference-count":19,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2023,1,22]],"date-time":"2023-01-22T00:00:00Z","timestamp":1674345600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["62172116"],"award-info":[{"award-number":["62172116"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["61972109"],"award-info":[{"award-number":["61972109"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["20200115-9"],"award-info":[{"award-number":["20200115-9"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["2674"],"award-info":[{"award-number":["2674"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Guangzhou Academician and Expert Workstation","award":["62172116"],"award-info":[{"award-number":["62172116"]}]},{"name":"Guangzhou Academician and Expert Workstation","award":["61972109"],"award-info":[{"award-number":["61972109"]}]},{"name":"Guangzhou Academician and Expert Workstation","award":["20200115-9"],"award-info":[{"award-number":["20200115-9"]}]},{"name":"Guangzhou Academician and Expert Workstation","award":["2674"],"award-info":[{"award-number":["2674"]}]},{"name":"Deanship of Scientific Research, Vice Presidency for Graduate Studies and Scientific Research, King Faisal University, Saudi Arabia","award":["62172116"],"award-info":[{"award-number":["62172116"]}]},{"name":"Deanship of Scientific Research, Vice Presidency for Graduate Studies and Scientific Research, King Faisal University, Saudi Arabia","award":["61972109"],"award-info":[{"award-number":["61972109"]}]},{"name":"Deanship of Scientific Research, Vice Presidency for Graduate Studies and Scientific Research, King Faisal University, Saudi Arabia","award":["20200115-9"],"award-info":[{"award-number":["20200115-9"]}]},{"name":"Deanship of Scientific Research, Vice Presidency for Graduate Studies and Scientific Research, King Faisal University, Saudi Arabia","award":["2674"],"award-info":[{"award-number":["2674"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>In mathematical chemistry, molecular descriptors serve an important role, primarily in quantitative structure\u2013property relationship (QSPR) and quantitative structure\u2013activity relationship (QSAR) studies. A topological index of a molecular graph is a real number that is invariant under graph isomorphism conditions and provides information about its size, symmetry, degree of branching, and cyclicity. For any graph N, the first and second Gourava indices are defined as GO1(N)=\u2211u\u2032v\u2032\u2208E(N)(d(u\u2032)+d(v\u2032)+d(u\u2032)d(v\u2032)) and GO2(N)=\u2211u\u2032v\u2032\u2208E(N)(d(u\u2032)+d(v\u2032))d(u\u2032)d(v\u2032), respectively.The independence number of a graph N, being the cardinality of its maximal independent set, plays a vital role in reading the energies of chemical trees. In this research paper, it is shown that among the family of trees of order \u03b4 and independence number \u03be, the spur tree denoted as \u03a5\u03b4,\u03be maximizes both 1st and 2nd Gourava indices, and with these characterizations this graph is unique.<\/jats:p>","DOI":"10.3390\/sym15020308","type":"journal-article","created":{"date-parts":[[2023,1,23]],"date-time":"2023-01-23T03:26:41Z","timestamp":1674444401000},"page":"308","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":["On Trees with Given Independence Numbers with Maximum Gourava Indices"],"prefix":"10.3390","volume":"15","author":[{"given":"Ying","family":"Wang","sequence":"first","affiliation":[{"name":"Software Engineering Institute of Guangzhou, Guangzhou 510980, China"},{"name":"Institute of Computing Science and Technology, Guangzhou University, Guangzhou 510006, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4523-8023","authenticated-orcid":false,"given":"Adnan","family":"Aslam","sequence":"additional","affiliation":[{"name":"Department of Natural Sciences and Humanities, University of Engineering and Technology, Lahore 54000, Pakistan"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2578-9816","authenticated-orcid":false,"given":"Nazeran","family":"Idrees","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Government College University, Faisalabad 37000, Pakistan"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7560-5317","authenticated-orcid":false,"given":"Salma","family":"Kanwal","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Lahore College for Women University, Lahore 54000, Pakistan"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Nabeela","family":"Iram","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Lahore College for Women University, Lahore 54000, Pakistan"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7579-8972","authenticated-orcid":false,"given":"Asima","family":"Razzaque","sequence":"additional","affiliation":[{"name":"Department of Basic Science, King Faisal University, Al-Ahsa 31982, Saudi Arabia"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2023,1,22]]},"reference":[{"key":"ref_1","unstructured":"West, D.B. (1996). Introduction to Graph Theory, Prentice Hall."},{"key":"ref_2","first-page":"715","article-title":"Maximum general sum-connectivity index for trees with given independence number","volume":"72","author":"Tomescu","year":"2014","journal-title":"MATCH Commun. Math. Comput. Chem."},{"key":"ref_3","first-page":"301","article-title":"On Zagreb and harary indices","volume":"70","author":"Das","year":"2013","journal-title":"MATCH Commun. Math. Comput. Chem."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"1127","DOI":"10.1021\/cr0780006","article-title":"Some new trends in chemical graph theory","volume":"108","author":"Pogliani","year":"2008","journal-title":"Chem. Rev."},{"key":"ref_5","unstructured":"Todeschini, R., and Consonni, V. (2008). Handbook of Molecular Descriptors, John Wiley & Sons."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"478","DOI":"10.1016\/0009-2614(93)87094-J","article-title":"Novel molecular descriptor for structure\u2014Property studies","volume":"211","year":"1993","journal-title":"Chem. Phys. Lett."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"33","DOI":"10.22457\/apam.v14n1a4","article-title":"The Gourava indices and coindices of graphs","volume":"14","author":"Kulli","year":"2017","journal-title":"Ann. Pure Appl. Math."},{"key":"ref_8","first-page":"235","article-title":"The product connectivity Gourava index","volume":"8","author":"Kulli","year":"2017","journal-title":"J. Comput. Math. Sci."},{"key":"ref_9","first-page":"116","article-title":"On Hyper Gourava indices and coindices","volume":"8","author":"Kulli","year":"2017","journal-title":"Int. J. Math. Arch."},{"key":"ref_10","first-page":"775","article-title":"Trees of Given Order and Independence Number with Minimal First Zagreb Index","volume":"72","author":"Vasilyev","year":"2014","journal-title":"MATCH Commun. Math. Comput. Chem."},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Das, J. (2022). On extremal Sombor index of trees with a given independence number \u03b1. arXiv.","DOI":"10.1016\/j.amc.2021.126575"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"6602899","DOI":"10.1155\/2022\/6602899","article-title":"On Acyclic Structures with Greatest First Gourava Invariant","volume":"2022","author":"Imtiaz","year":"2022","journal-title":"J. Chem."},{"key":"ref_13","first-page":"297","article-title":"Gourava and Hyper-Gourava indices of some Cactus Chains","volume":"27","author":"Basavanagoud","year":"2021","journal-title":"J. Indones. Math. Soc."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"7513770","DOI":"10.1155\/2022\/7513770","article-title":"On Second Gourava Invariant for q-Apex Trees","volume":"2022","author":"Wang","year":"2022","journal-title":"J. Chem."},{"key":"ref_15","first-page":"8291974","article-title":"On Trees with Greatest Invariant Using Edge Swapping Operation","volume":"2022","author":"Wang","year":"2022","journal-title":"Comput. Intell. Neurosci."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"90","DOI":"10.1016\/j.dam.2015.10.010","article-title":"On the structural properties of trees with minimal atom-bond connectivity index III: Bounds on B1- and B2-branches","volume":"204","author":"Dimitrov","year":"2016","journal-title":"Discret. Appl. Math."},{"key":"ref_17","first-page":"141","article-title":"The smallest ABC index of trees with n pendant vertices","volume":"76","author":"Gao","year":"2016","journal-title":"MATCH Commun. Math. Comput. Chem."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"3581","DOI":"10.2298\/FIL1810581H","article-title":"More results on extremum Randi\u0107 indices of (molecular) trees","volume":"32","author":"Husin","year":"2018","journal-title":"Filomat"},{"key":"ref_19","first-page":"409","article-title":"The Randi\u0107 indices of trees, unicyclic graphs and bicyclic graphs","volume":"127","author":"Li","year":"2016","journal-title":"Ars Combin."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/15\/2\/308\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T18:13:13Z","timestamp":1760119993000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/15\/2\/308"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,1,22]]},"references-count":19,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2023,2]]}},"alternative-id":["sym15020308"],"URL":"https:\/\/doi.org\/10.3390\/sym15020308","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,1,22]]}}}