{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T00:40:12Z","timestamp":1760143212297,"version":"build-2065373602"},"reference-count":44,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2024,1,18]],"date-time":"2024-01-18T00:00:00Z","timestamp":1705536000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Institutional Fund Projects","award":["IFPDP-151-22"],"award-info":[{"award-number":["IFPDP-151-22"]}]},{"name":"Ministry of Education and Deanship of Scientific Research (DSR), King Abdulaziz University (KAU), Jeddah, Saudi Arabia","award":["IFPDP-151-22"],"award-info":[{"award-number":["IFPDP-151-22"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>In this work, we determine the maximum general Randi\u0107 index (a general symmetric function of vertex degrees) for \u03b70\u2264\u03b7&lt;0 among all n-vertex unicyclic graphs with a fixed maximum degree \u0394 and the maximum and the second maximum general Randi\u0107 index for \u03b70\u2264\u03b7&lt;0 among all n-vertex unicyclic graphs, where \u03b70\u2248\u22120.21. We establish sharp inequalities and identify the graphs attaining the inequalities. Thereby, extremal graphs are obtained for the general Randi\u0107 index, and certain open gaps in the theory of extremal unicyclic graphs are filled (some open problems are provided). We use computational software to calculate the Randi\u0107 index for the chemical trees up to order 7 and use the statistical (linear regression) analysis to discuss the various applications of the Randi\u0107 index with the physical properties of drugs on the said chemical trees. We show that the Randi\u0107 index is better correlated with the heat of vaporization for these alkanes.<\/jats:p>","DOI":"10.3390\/sym16010113","type":"journal-article","created":{"date-parts":[[2024,1,18]],"date-time":"2024-01-18T03:50:56Z","timestamp":1705549856000},"page":"113","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["General Randi\u0107 Index of Unicyclic Graphs and Its Applications to Drugs"],"prefix":"10.3390","volume":"16","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-2729-7396","authenticated-orcid":false,"given":"Alaa","family":"Altassan","sequence":"first","affiliation":[{"name":"Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2827-0462","authenticated-orcid":false,"given":"Muhammad","family":"Imran","sequence":"additional","affiliation":[{"name":"Department of Mathematical Sciences, United Arab Emirates University, Al Ain 15551, United Arab Emirates"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2024,1,18]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"6609","DOI":"10.1021\/ja00856a001","article-title":"On characterization of molecular branching","volume":"97","year":"1975","journal-title":"J. Am. Chem. Soc."},{"key":"ref_2","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. 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