{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,1]],"date-time":"2026-03-01T09:58:10Z","timestamp":1772359090066,"version":"3.50.1"},"reference-count":29,"publisher":"Wiley","issue":"1","license":[{"start":{"date-parts":[[2021,6,24]],"date-time":"2021-06-24T00:00:00Z","timestamp":1624492800000},"content-version":"vor","delay-in-days":174,"URL":"http:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Research and Innovation, Ministry of Education in Saudi Arabia","award":["IFP-2020-17"],"award-info":[{"award-number":["IFP-2020-17"]}]}],"content-domain":{"domain":["onlinelibrary.wiley.com"],"crossmark-restriction":true},"short-container-title":["Complexity"],"published-print":{"date-parts":[[2021,1]]},"abstract":"<jats:p>Graph theory is a dynamic tool for designing and modeling of an interconnection system by a graph. The vertices of such graph are processor nodes and edges are the connections between these processors nodes. The topology of a system decides its best use. Geometric\u2010arithmetic index is one of the most studied graph invariant to characterize the topological aspects of underlying interconnection networks or graphs. Transformation over graph is also an important tool to define new network of their own choice in computer science. In this work, we discuss transformed family of graphs. Let  be the connected graph comprises by <jats:italic>k<\/jats:italic> number of pendent path attached with fully connected vertices of the <jats:italic>n<\/jats:italic>\u2010vertex connected graph \u0393. Let  and  be the transformed graphs under the fact of transformations <jats:italic>A<\/jats:italic><jats:sub><jats:italic>\u03b1<\/jats:italic><\/jats:sub> and , 0 \u2264 <jats:italic>\u03b1<\/jats:italic> \u2264 <jats:italic>l<\/jats:italic>, 0 \u2264 <jats:italic>\u03b2<\/jats:italic> \u2264 <jats:italic>k<\/jats:italic> \u2212 1, respectively. In this work, we obtained new inequalities for the graph family  and transformed graphs  and  which involve GA(\u0393). The presence of GA(\u0393) makes the inequalities more general than all those which were previously defined for the GA index. Furthermore, we characterize extremal graphs which make the inequalities tight.<\/jats:p>","DOI":"10.1155\/2021\/3745862","type":"journal-article","created":{"date-parts":[[2021,6,24]],"date-time":"2021-06-24T23:05:31Z","timestamp":1624575931000},"update-policy":"https:\/\/doi.org\/10.1002\/crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":["An Approach to the Geometric\u2010Arithmetic Index for Graphs under Transformations\u2019 Fact over Pendent Paths"],"prefix":"10.1155","volume":"2021","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-8669-942X","authenticated-orcid":false,"given":"Muhammad","family":"Asif","sequence":"first","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8328-2979","authenticated-orcid":false,"given":"Hamad","family":"Almohamedh","sequence":"additional","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1768-3111","authenticated-orcid":false,"given":"Muhammad","family":"Hussain","sequence":"additional","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6048-3724","authenticated-orcid":false,"given":"Khalid M","family":"Alhamed","sequence":"additional","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4754-5019","authenticated-orcid":false,"given":"Abdulrazaq A.","family":"Almutairi","sequence":"additional","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2050-5236","authenticated-orcid":false,"given":"Sultan","family":"Almotairi","sequence":"additional","affiliation":[]}],"member":"311","published-online":{"date-parts":[[2021,6,24]]},"reference":[{"key":"e_1_2_9_1_2","doi-asserted-by":"publisher","DOI":"10.1109\/2.19830"},{"key":"e_1_2_9_2_2","doi-asserted-by":"publisher","DOI":"10.1109\/c-m.1981.220290"},{"key":"e_1_2_9_3_2","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-349-03521-2"},{"key":"e_1_2_9_4_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.amc.2014.07.064"},{"key":"e_1_2_9_5_2","doi-asserted-by":"publisher","DOI":"10.1139\/cjc-2015-0185"},{"key":"e_1_2_9_6_2","doi-asserted-by":"publisher","DOI":"10.1155\/2020\/3794592"},{"key":"e_1_2_9_7_2","doi-asserted-by":"publisher","DOI":"10.1201\/9781315139111"},{"key":"e_1_2_9_8_2","doi-asserted-by":"publisher","DOI":"10.1155\/2018\/5729291"},{"key":"e_1_2_9_9_2","doi-asserted-by":"publisher","DOI":"10.1155\/2021\/5539904"},{"key":"e_1_2_9_10_2","doi-asserted-by":"publisher","DOI":"10.1023\/a:1010767517079"},{"key":"e_1_2_9_11_2","doi-asserted-by":"publisher","DOI":"10.1021\/ci9900175"},{"key":"e_1_2_9_12_2","doi-asserted-by":"publisher","DOI":"10.1007\/s40747-020-00262-0"},{"key":"e_1_2_9_13_2","doi-asserted-by":"publisher","DOI":"10.1002\/mma.6896"},{"key":"e_1_2_9_14_2","doi-asserted-by":"publisher","DOI":"10.1021\/ja01193a005"},{"key":"e_1_2_9_15_2","doi-asserted-by":"publisher","DOI":"10.1021\/ja00856a001"},{"key":"e_1_2_9_16_2","doi-asserted-by":"publisher","DOI":"10.1007\/s10910-009-9520-x"},{"key":"e_1_2_9_17_2","doi-asserted-by":"publisher","DOI":"10.1007\/s10910-009-9603-8"},{"key":"e_1_2_9_18_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.cplett.2010.07.097"},{"key":"e_1_2_9_19_2","first-page":"691","article-title":"A unified approach to extremal trees with respect to geometric-arithmetic, Szeged and edge Szeged indices","volume":"65","author":"Hua H.","year":"2011","journal-title":"MATCH Communications in Mathematical and in Computer Chemistry"},{"key":"e_1_2_9_20_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.dam.2013.08.001"},{"key":"e_1_2_9_21_2","first-page":"103","article-title":"On the geometric\u2013arithmetic index","volume":"74","author":"Rodr\u0131guez J.","year":"2015","journal-title":"MATCH Communications in Mathematical and in Computer Chemistry"},{"key":"e_1_2_9_22_2","doi-asserted-by":"publisher","DOI":"10.18514\/mmn.2015.1423"},{"key":"e_1_2_9_23_2","first-page":"59","article-title":"Recent lower bounds for geometric-arithmetic index","volume":"1","author":"Portilla A.","year":"2019","journal-title":"Discrete Mathematics Letters"},{"key":"e_1_2_9_24_2","first-page":"8","article-title":"Bicyclic graphs with maximum geometric-arithmetic index","volume":"20","author":"Hasni R.","year":"2020","journal-title":"Applied Mathematics E-Notes"},{"key":"e_1_2_9_25_2","first-page":"619","article-title":"On geometric\u2013arithmetic index of graphs","volume":"64","author":"Das K. 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