{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,9,30]],"date-time":"2026-09-30T21:10:16Z","timestamp":1790802616225,"version":"4.1.0"},"reference-count":54,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2018,10,3]],"date-time":"2018-10-03T00:00:00Z","timestamp":1538524800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Biomolecules"],"abstract":"<jats:p>Topological indices are numerical parameters used to study the physical and chemical properties of compounds. In quantitative structure\u2013activity relationship QSARs, topological indices correlate the biological activity of compounds with their physical properties like boiling point, stability, melting point, distortion, and strain energy etc. In this paper, we determined the M-polynomials of the crystallographic structure of the molecules Cu2O and TiF2 [p,q,r]. Then we derived closed formulas for some well-known topological indices using calculus. In the end, we used Maple 15 to plot surfaces associated with the topological indices of Cu2O and TiF2 [p,q,r].<\/jats:p>","DOI":"10.3390\/biom8040107","type":"journal-article","created":{"date-parts":[[2018,10,4]],"date-time":"2018-10-04T02:19:49Z","timestamp":1538619589000},"page":"107","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":35,"title":["M-Polynomials and Degree-Based Topological Indices of the Crystallographic Structure of Molecules"],"prefix":"10.3390","volume":"8","author":[{"given":"Wei","family":"Gao","sequence":"first","affiliation":[{"name":"School of Information Science and Technology, Yunnan Normal University, Kunming 650500, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Muhammad","family":"Younas","sequence":"additional","affiliation":[{"name":"Department of Mathematics, COMSATS University Islamabad, Lahore Campus, Lahore 54000, Pakistan"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Adeel","family":"Farooq","sequence":"additional","affiliation":[{"name":"Department of Mathematics, COMSATS University Islamabad, Lahore Campus, Lahore 54000, Pakistan"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Abid","family":"Mahboob","sequence":"additional","affiliation":[{"name":"Department of Mathematics, University of Education, Vehari Campus, Lahore 54000, Pakistan"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5488-0467","authenticated-orcid":false,"given":"Waqas","family":"Nazeer","sequence":"additional","affiliation":[{"name":"Division of Science and Technology, University of Education, Lahore 54000, Pakistan"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2018,10,3]]},"reference":[{"key":"ref_1","first-page":"299","article-title":"PI polynomial of zig\u2014zag polyhex nanotubes","volume":"3","author":"Loghman","year":"2008","journal-title":"Dig. 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