{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,29]],"date-time":"2025-10-29T19:44:36Z","timestamp":1761767076378,"version":"build-2065373602"},"reference-count":48,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2022,6,19]],"date-time":"2022-06-19T00:00:00Z","timestamp":1655596800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Umm Al-Qura University","award":["22UQU4310392DSR17"],"award-info":[{"award-number":["22UQU4310392DSR17"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>A topological index is a number generated from a molecular structure (i.e., a graph) that indicates the essential structural properties of the proposed molecule. Indeed, it is an algebraic quantity connected with the chemical structure that correlates it with various physical characteristics. It is possible to determine several different properties, such as chemical activity, thermodynamic properties, physicochemical activity, and biological activity, using several topological indices, such as the geometric-arithmetic index, arithmetic-geometric index, Randi\u0107 index, and the atom-bond connectivity indices. Consider G as a group and H as a non-empty subset of G. The commuting graph C(G,H), has H as the vertex set, where h1,h2\u2208H are edge connected whenever h1 and h2 commute in G. This article examines the topological characteristics of commuting graphs having an algebraic structure by computing their atomic-bond connectivity index, the Wiener index and its reciprocal, the harmonic index, geometric-arithmetic index, Randi\u0107 index, Harary index, and the Schultz molecular topological index. Moreover, we study the Hosoya properties, such as the Hosoya polynomial and the reciprocal statuses of the Hosoya polynomial of the commuting graphs of finite subgroups of SL(2,C). Finally, we compute the Z-index of the commuting graphs of the binary dihedral groups.<\/jats:p>","DOI":"10.3390\/sym14061266","type":"journal-article","created":{"date-parts":[[2022,6,19]],"date-time":"2022-06-19T21:19:26Z","timestamp":1655673566000},"page":"1266","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":18,"title":["On the Topological Indices of Commuting Graphs for Finite Non-Abelian Groups"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-7788-791X","authenticated-orcid":false,"given":"Fawad","family":"Ali","sequence":"first","affiliation":[{"name":"Institute of Numerical Sciences, Kohat University of Science & Technology, Kohat 26000, Khyber Pakhtunkhwa, Pakistan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1381-0291","authenticated-orcid":false,"given":"Bilal A.","family":"Rather","sequence":"additional","affiliation":[{"name":"Mathematical Sciences Department, College of Science, United Arab Emirate University, Al Ain 15551, Abu Dhabi, United Arab Emirates"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1511-9760","authenticated-orcid":false,"given":"Nahid","family":"Fatima","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Sciences, Prince Sultan University, Riyadh 11586, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4978-9287","authenticated-orcid":false,"given":"Muhammad","family":"Sarfraz","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Sun Yat-sen University, Guangzhou 510275, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4361-7786","authenticated-orcid":false,"given":"Asad","family":"Ullah","sequence":"additional","affiliation":[{"name":"Department of Mathematical Sciences, University of Lakki Marwat, Lakki Marwat 28420, Khyber Pakhtunkhwa, Pakistan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7290-8398","authenticated-orcid":false,"given":"Khalid Abdulkhaliq M.","family":"Alharbi","sequence":"additional","affiliation":[{"name":"Mechanical Engineering Department, College of Engineering, Umm Al-Qura University, Makkah 24382, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Rahim","family":"Dad","sequence":"additional","affiliation":[{"name":"Department of Mathematics, University of Science and Technology Bannu, Bannu 28100, Khyber Pakhtunkhwa, Pakistan"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2022,6,19]]},"reference":[{"key":"ref_1","first-page":"1419","article-title":"Computing ABC4 index of nanostar dendrimers","volume":"4","author":"Ghorbani","year":"2010","journal-title":"Optoelectron. 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