{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T02:28:07Z","timestamp":1760236087011,"version":"build-2065373602"},"reference-count":30,"publisher":"MDPI AG","issue":"11","license":[{"start":{"date-parts":[[2021,10,21]],"date-time":"2021-10-21T00:00:00Z","timestamp":1634774400000},"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>Graph theory can be used to optimize interconnection network systems. The compatibility of such networks mainly depends on their topology. Topological indices may characterize the topology of such networks. In this work, we studied a symmetric network \u03b8\u03d5 formed by \u03d5 time repetition of the process of joining \u03b8 copies of a selected graph \u03a9 in such a way that corresponding vertices of \u03a9 in all the copies are joined with each other by a new edge. The symmetry of \u03b8\u03d5 is ensured by the involvement of complete graph K\u03b8 in the construction process. The free hand to choose an initial graph \u03a9 and formation of chemical graphs using \u03b8\u03d5\u03a9 enhance its importance as a family of graphs which covers all the pre-defined graphs, along with space for new graphs, possibly formed in this way. We used Zagreb connection indices for the characterization of \u03b8\u03d5\u03a9. These indices have gained worth in the field of chemical graph theory in very small duration due to their predictive power for enthalpy, entropy, and acentric factor. These computations are mathematically novel and assist in topological characterization of \u03b8\u03d5\u03a9 to enable its emerging use.<\/jats:p>","DOI":"10.3390\/sym13111991","type":"journal-article","created":{"date-parts":[[2021,10,21]],"date-time":"2021-10-21T23:27:39Z","timestamp":1634858859000},"page":"1991","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["Study of \u03b8\u03d5 Networks via Zagreb Connection Indices"],"prefix":"10.3390","volume":"13","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-8669-942X","authenticated-orcid":false,"given":"Muhammad","family":"Asif","sequence":"first","affiliation":[{"name":"Department of Mathematics, COMSATS University Islamabad (Lahore Campus), Lahore 53713, Pakistan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5736-4014","authenticated-orcid":false,"given":"Bart\u0142omiej","family":"Kizielewicz","sequence":"additional","affiliation":[{"name":"Research Team on Intelligent Decision Support Systems, Department of Artificial Intelligence and Applied Mathematics, Faculty of Computer Science and Information Technology, West Pomeranian University of Technology in Szczecin, ul. \u017bo\u0142nierska 49, 71-210 Szczecin, Poland"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3199-3346","authenticated-orcid":false,"given":"Atiq ur","family":"Rehman","sequence":"additional","affiliation":[{"name":"Department of Mathematics, COMSATS University Islamabad (Lahore Campus), Lahore 53713, Pakistan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1768-3111","authenticated-orcid":false,"given":"Muhammad","family":"Hussain","sequence":"additional","affiliation":[{"name":"Department of Mathematics, COMSATS University Islamabad (Lahore Campus), Lahore 53713, Pakistan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7076-2519","authenticated-orcid":false,"given":"Wojciech","family":"Sa\u0142abun","sequence":"additional","affiliation":[{"name":"Research Team on Intelligent Decision Support Systems, Department of Artificial Intelligence and Applied Mathematics, Faculty of Computer Science and Information Technology, West Pomeranian University of Technology in Szczecin, ul. \u017bo\u0142nierska 49, 71-210 Szczecin, Poland"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2021,10,21]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"12","DOI":"10.1109\/C-M.1981.220290","article-title":"A survey of interconnection networks","volume":"14","author":"Feng","year":"1981","journal-title":"Computer"},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Bondy, J.A., and Murty, U.S.R. 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