{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T01:41:48Z","timestamp":1760146908327,"version":"build-2065373602"},"reference-count":30,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2024,12,30]],"date-time":"2024-12-30T00:00:00Z","timestamp":1735516800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Scientific Research Deanship at the University of Ha\u2019il, Saudi Arabia","award":["RG-24 046"],"award-info":[{"award-number":["RG-24 046"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>This paper is mainly concerned with the study of two bond incident degree (BID) indices, namely the variable sum exdeg index SEIa and the general zeroth-order Randi\u0107 index R\u03b10. The minimum values of SEIa and R\u03b10 in the class of all trees of fixed order containing no vertex of even degree are obtained for a&gt;1 and \u03b1\u2208[0,1]; also, the maximum value of R\u03b10 in the mentioned class is determined for 0&lt;\u03b1&lt;1. Moreover, in the family of all trees of fixed order and with a given number of vertices of even degrees, the extremum values of SEIa and R\u03b10 are found for every real number \u03b1\u2209{0,1} and a&gt;1. Furthermore, in the class of all trees of fixed order and with a given number of vertices of maximum degree, the minimum values of SEIa and R\u03b10 are determined when a&gt;1 and \u03b1 does not belong to the closed interval [0,1]; in the same class, the maximum values of R\u03b10 are also found for 0&lt;\u03b1&lt;1. The graphs that achieve the obtained extremal values are also determined.<\/jats:p>","DOI":"10.3390\/axioms14010023","type":"journal-article","created":{"date-parts":[[2024,12,31]],"date-time":"2024-12-31T07:51:57Z","timestamp":1735631517000},"page":"23","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["On Trees with a Given Number of Vertices of Fixed Degree and Their Two Bond Incident Degree Indices"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-7619-6869","authenticated-orcid":false,"given":"Abeer M.","family":"Albalahi","sequence":"first","affiliation":[{"name":"Department of Mathematics, College of Science, University of Ha\u2019il, Ha\u2019il P.O. Box 2240, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Muhammad","family":"Rizwan","sequence":"additional","affiliation":[{"name":"Department of Sciences and Humanities, National University of Computer and Emerging Sciences, B-Block, Faisal Town, Lahore 54770, Pakistan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Akhlaq A.","family":"Bhatti","sequence":"additional","affiliation":[{"name":"Department of Sciences and Humanities, National University of Computer and Emerging Sciences, B-Block, Faisal Town, Lahore 54770, Pakistan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9681-1550","authenticated-orcid":false,"given":"Ivan","family":"Gutman","sequence":"additional","affiliation":[{"name":"Faculty of Science, University of Kragujevac, 34000 Kragujevac, Serbia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8160-4196","authenticated-orcid":false,"given":"Akbar","family":"Ali","sequence":"additional","affiliation":[{"name":"Department of Mathematics, College of Science, University of Ha\u2019il, Ha\u2019il P.O. Box 2240, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Tariq","family":"Alraqad","sequence":"additional","affiliation":[{"name":"Department of Mathematics, College of Science, University of Ha\u2019il, Ha\u2019il P.O. Box 2240, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Hicham","family":"Saber","sequence":"additional","affiliation":[{"name":"Department of Mathematics, College of Science, University of Ha\u2019il, Ha\u2019il P.O. Box 2240, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2024,12,30]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Bondy, J.A., and Murty, U.S.R. (2008). Graph Theory, Springer.","DOI":"10.1007\/978-1-84628-970-5"},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Gross, J.L., Yellen, J., and Zhang, P. (2013). Section 13.1. Chemical Graph Theory. 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