{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T01:24:53Z","timestamp":1760059493795,"version":"build-2065373602"},"reference-count":21,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2025,6,18]],"date-time":"2025-06-18T00:00:00Z","timestamp":1750204800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>The field related to indices was developed by researchers for various purposes. Optimization is one of the purposes used by researchers in different situations. In this article, a generalized Sombor index is considered. This work is related to the idea of optimization in the families of bicyclic graphs, trees, and unicyclic graphs. We investigated optimal values in the stated families by means of well-known transformations. The transformations include the following: Transformation A, Transformation B, Transformation C, and Transformation D. Transformation A and Transformation B increase the value of the generalized Sombor index, while Transformation C and Transformation D are used for minimal values.<\/jats:p>","DOI":"10.3390\/sym17060968","type":"journal-article","created":{"date-parts":[[2025,6,18]],"date-time":"2025-06-18T05:30:42Z","timestamp":1750224642000},"page":"968","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Investigation of General Sombor Index for Optimal Values in Bicyclic Graphs, Trees, and Unicyclic Graphs Using Well-Known Transformations"],"prefix":"10.3390","volume":"17","author":[{"ORCID":"https:\/\/orcid.org\/0009-0006-5447-8992","authenticated-orcid":false,"given":"Miraj","family":"Khan","sequence":"first","affiliation":[{"name":"Department of Mathematics, Islamia College Peshawar, Peshawar 25120, Khyber Pakhtunkhwa, Pakistan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3859-0205","authenticated-orcid":false,"given":"Muhammad Yasin","family":"Khan","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Islamia College Peshawar, Peshawar 25120, Khyber Pakhtunkhwa, Pakistan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0009-0008-4444-0370","authenticated-orcid":false,"given":"Gohar","family":"Ali","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Islamia College Peshawar, Peshawar 25120, Khyber Pakhtunkhwa, Pakistan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8042-1806","authenticated-orcid":false,"given":"Ioan-Lucian","family":"Popa","sequence":"additional","affiliation":[{"name":"Department of Computing, Mathematics and Electronics, 1 Decembrie 1918 University of Alba Iulia, 510009 Alba Iulia, Romania"},{"name":"Faculty of Mathematics and Computer Science, Transilvania University of Brasov, Iuliu Maniu Street 50, 500091 Brasov, Romania"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2025,6,18]]},"reference":[{"key":"ref_1","unstructured":"Balakrishnan, V.K. (1997). Schaum\u2019s Outline of Theory and Problems of Graph Theory, McGraw-Hill."},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Khan, M.Y., Ali, G., and Popa, I.L. (2024). Optimization of General Power-Sum Connectivity Index in Uni-Cyclic Graphs, Bi-Cyclic Graphs and Trees by Means of Operations. Axioms, 13.","DOI":"10.3390\/axioms13120840"},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Khan, M.Y., Ali, G., and Popa, I.L. (2025). Optimization in Symmetric Trees, Unicyclic Graphs, and Bicyclic Graphs with Help of Mappings Using Second Form of Generalized Power-Sum Connectivity Index. Symmetry, 17.","DOI":"10.3390\/sym17010122"},{"key":"ref_4","unstructured":"Diudea, M.V., Gutman, I., and Jantschi, L. (2001). Molecular Topology, Nova Science Publishers."},{"key":"ref_5","first-page":"3139867","article-title":"Optimizing Wiener and Randi\u0107 Indices of Graphs","volume":"2020","author":"Mahasinghe","year":"2020","journal-title":"Adv. Oper. Res."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"982","DOI":"10.1021\/ci960007t","article-title":"The quasi-Wiener and the Kirchhoff indices coincide","volume":"36","author":"Gutman","year":"1996","journal-title":"J. Chem. Inf. Comput. Sci."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"441","DOI":"10.1177\/016555150202800601","article-title":"Social network analysis: A powerful strategy, also for the information sciences","volume":"28","author":"Otte","year":"2002","journal-title":"J. Inf. Sci."},{"key":"ref_8","first-page":"936","article-title":"On topological indices of certain interconnection networks","volume":"244","author":"Imran","year":"2014","journal-title":"Appl. Math. Comput."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"417","DOI":"10.1002\/med.2610070404","article-title":"Indexes of molecular shape from chemical graphs","volume":"7","author":"Kier","year":"1987","journal-title":"Med. Res. Rev."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"1884","DOI":"10.1021\/ie980682n","article-title":"Optimization in polymer design using connectivity indices","volume":"38","author":"Camarda","year":"1999","journal-title":"Ind. Eng. Chem. Res."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"343","DOI":"10.1016\/j.cplett.2005.05.096","article-title":"Generalised topological indices: Optimisation methodology and physico-chemical interpretation","volume":"410","author":"Matamala","year":"2005","journal-title":"Chem. Phys. Lett."},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Preu\u00df, M., Dehmer, M., Pickl, S., and Holzinger, A. (2014). On terrain coverage optimization by using a network approach for universal graph-based data mining and knowledge discovery. Brain Informatics and Health, Proceedings of the International Conference, BIH 2014, Warsaw, Poland, 11\u201314 August 2014, Springer International Publishing. Proceedings.","DOI":"10.1007\/978-3-319-09891-3_51"},{"key":"ref_13","first-page":"635","article-title":"Lower and upper bounds of the forgotten topological index","volume":"76","author":"Che","year":"2016","journal-title":"MATCH Commun. Math. Comput. Chem."},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Monsalve, J., and Rada, J. (2021). Sharp upper and lower bounds of VDB topological indices of digraphs. Symmetry, 13.","DOI":"10.3390\/sym13101903"},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Mart\u00ednez-P\u00e9rez, \u00c1., and Rodr\u00edguez, J.M. (2020). New bounds for topological indices on trees through generalized methods. Symmetry, 12.","DOI":"10.3390\/sym12071097"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"108591","DOI":"10.1016\/j.topol.2023.108591","article-title":"M Upper and lower bounds for topological indices on unicyclic graphs","volume":"339","year":"2023","journal-title":"Topol. Its Appl."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"6289518","DOI":"10.1155\/2020\/6289518","article-title":"Some topological invariants of graphs associated with the group of symmetries","volume":"2020","author":"Wei","year":"2020","journal-title":"J. Chem."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"105845","DOI":"10.1016\/j.enganabound.2024.105845","article-title":"A group theory based topology optimization scheme for the design of inhomogeneous waveguides with dihedral group symmetries","volume":"166","author":"Chu","year":"2024","journal-title":"Eng. Anal. Bound. Elem."},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"\u00d6zt\u00fcrk S\u00f6zen, E., Alsuraiheed, T., Abdio\u011flu, C., and Ali, S. (2023). Computing topological descriptors of prime ideal sum graphs of commutative rings. Symmetry, 15.","DOI":"10.3390\/sym15122133"},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Balasubramanian, K. (2023). Topological indices, graph spectra, entropies, Laplacians, and matching polynomials of n-dimensional hypercubes. Symmetry, 15.","DOI":"10.3390\/sym15020557"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"47","DOI":"10.1016\/S0045-7949(98)00158-8","article-title":"A symmetry reduction method for continuum structural topology optimization","volume":"70","author":"Kosaka","year":"1999","journal-title":"Comput. Struct."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/17\/6\/968\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,9]],"date-time":"2025-10-09T17:54:00Z","timestamp":1760032440000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/17\/6\/968"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,6,18]]},"references-count":21,"journal-issue":{"issue":"6","published-online":{"date-parts":[[2025,6]]}},"alternative-id":["sym17060968"],"URL":"https:\/\/doi.org\/10.3390\/sym17060968","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2025,6,18]]}}}