{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,9]],"date-time":"2025-10-09T00:39:22Z","timestamp":1759970362675,"version":"build-2065373602"},"reference-count":18,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2025,1,15]],"date-time":"2025-01-15T00:00:00Z","timestamp":1736899200000},"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 topological index (TI), sometimes referred to as the connectivity index, is a molecular descriptor calculated based on the molecular graph of a chemical compound. Topological indices (TIs) are numeric parameters of a graph used to characterize its topology and are usually graph-invariant. The generalized power-sum connectivity index (GPSCI) for the graph is \u03a9Y\u03b1(\u03a9)=\u2211\u03b6\u03f1\u2208E(\u03a9)(d\u03a9(\u03b6)d\u03a9(\u03b6)+d\u03a9(\u03f1)d\u03a9(\u03f1))\u03b1, while the second form of the GPSCI is defined as Y\u03b2(\u03a9)=\u2211\u03b6\u03f1\u2208E(\u03a9)(d\u03a9(\u03b6)d\u03a9(\u03b6)\u00d7d\u03a9(\u03f1)d\u03a9(\u03f1))\u03b2. In this paper, we investigate Y\u03b2 in the family of trees, unicyclic graphs, and bicyclic graphs. We determine optimal graphs in the desired families for Y\u03b2 using certain mappings. For graphs with maximal values, two mappings are used, namely A and B, while for graphs with minimal values, mapping C and mapping D are considered.<\/jats:p>","DOI":"10.3390\/sym17010122","type":"journal-article","created":{"date-parts":[[2025,1,15]],"date-time":"2025-01-15T04:42:08Z","timestamp":1736916128000},"page":"122","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Optimization in Symmetric Trees, Unicyclic Graphs, and Bicyclic Graphs with Help of Mappings Using Second Form of Generalized Power-Sum Connectivity Index"],"prefix":"10.3390","volume":"17","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-3859-0205","authenticated-orcid":false,"given":"Muhammad Yasin","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\/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":[[2025,1,15]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"17","DOI":"10.1021\/ja01193a005","article-title":"Structural determination of paraffin boiling points","volume":"69","author":"Wiener","year":"1947","journal-title":"J. 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