{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T02:58:59Z","timestamp":1760151539342,"version":"build-2065373602"},"reference-count":31,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2022,3,31]],"date-time":"2022-03-31T00:00:00Z","timestamp":1648684800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/2.zoppoz.workers.dev:443\/https\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100004281","name":"National Science Center","doi-asserted-by":"publisher","award":["UMO-2018\/29\/B\/HS4\/02725"],"award-info":[{"award-number":["UMO-2018\/29\/B\/HS4\/02725"]}],"id":[{"id":"10.13039\/501100004281","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Information"],"abstract":"<jats:p>A graph is a tool for designing a system\u2019s required interconnection network. The topology of such networks determines their compatibility. For the first time, in this work we construct subdivided \u03b7\u03b6 network S(\u03b7\u03b6\u0393) and discussed their topology. In graph theory, there are a variety of invariants to study the topology of a network, but topological indices are designed in such a way that these may transform the graph into a numeric value. In this work, we study S(\u03b7\u03b6\u0393) via Zagreb connection indices. Due to their predictive potential for enthalpy, entropy, and acentric factor, these indices gain value in the field of chemical graph theory in a very short time. \u03b7\u03b6\u0393 formed by \u03b6 time repeated process which consists \u03b7\u03b6 copies of graph \u0393 along with \u03b72|V(\u0393)|\u03b6\u03b7\u03b6\u22121 edges which used to join these \u03b7\u03b6 copies of \u0393. The free hand to choose the initial graph \u0393 for desired network S(\u03b7\u03b6\u0393) and its relation with chemical networks along with the repute of Zagreb connection indices enhance the worth of this study. These computations are theoretically innovative and aid topological characterization of S(\u03b7\u03b6\u0393).<\/jats:p>","DOI":"10.3390\/info13040179","type":"journal-article","created":{"date-parts":[[2022,3,31]],"date-time":"2022-03-31T21:34:29Z","timestamp":1648762469000},"page":"179","update-policy":"https:\/\/2.zoppoz.workers.dev:443\/https\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Study of Transformed \u03b7\u03b6 Networks via Zagreb Connection Indices"],"prefix":"10.3390","volume":"13","author":[{"ORCID":"https:\/\/2.zoppoz.workers.dev:443\/https\/orcid.org\/0000-0003-1768-3111","authenticated-orcid":false,"given":"Muhammad","family":"Hussain","sequence":"first","affiliation":[{"name":"Department of Mathematics, Lahore Campus, COMSATS University Islamabad, Lahore 53713, Pakistan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/2.zoppoz.workers.dev:443\/https\/orcid.org\/0000-0002-3199-3346","authenticated-orcid":false,"given":"Atiq ur","family":"Rehman","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Lahore Campus, COMSATS University Islamabad, Lahore 53713, Pakistan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/2.zoppoz.workers.dev:443\/https\/orcid.org\/0000-0002-0834-2019","authenticated-orcid":false,"given":"Andrii","family":"Shekhovtsov","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:\/\/2.zoppoz.workers.dev:443\/https\/orcid.org\/0000-0001-8669-942X","authenticated-orcid":false,"given":"Muhammad","family":"Asif","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Lahore Campus, COMSATS University Islamabad, Lahore 53713, Pakistan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/2.zoppoz.workers.dev:443\/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":[[2022,3,31]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Trinajstic, N. (2018). Chemical Graph Theory, Routledge.","DOI":"10.1201\/9781315139111"},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Liu, J.-B., Shaker, H., Nadeem, I., and Hussain, M. (2018). Topological aspects of Boron nanotubes. Adv. Mater. Sci. Eng., 2018.","DOI":"10.1155\/2018\/5729291"},{"key":"ref_3","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. Am. Chem. Soc."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"535","DOI":"10.1016\/0009-2614(72)85099-1","article-title":"Graph theory and molecular orbitals. Total \u03b6-electron energy of alternant hydrocarbons","volume":"17","author":"Gutman","year":"1972","journal-title":"Chem. Phys. Lett."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"37","DOI":"10.1007\/BF01046555","article-title":"A topological index for the total \u03a0-electron energy","volume":"38","author":"Hosoya","year":"1975","journal-title":"Theor. Chim. Acta"},{"key":"ref_6","first-page":"213","article-title":"The hyper-Zagreb index of graph operations","volume":"4","author":"Shirdel","year":"2013","journal-title":"Iran. J. Math. Chem."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"1800008","DOI":"10.1002\/minf.201800008","article-title":"A Novel\/Old Modification of the First Zagreb Index","volume":"37","author":"Ali","year":"2018","journal-title":"Mol. Inform."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"1800116","DOI":"10.1002\/minf.201800116","article-title":"Alkanes with the first three maximal\/minimal modified first Zagreb connection indices","volume":"38","author":"Du","year":"2019","journal-title":"Mol. Inform."},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Ali, U., Javaid, M., and Alanazi, A.M. (2020). Computing Analysis of Connection-Based Indices and Coindices for Product of Molecular Networks. Symmetry, 12.","DOI":"10.3390\/sym12081320"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"43","DOI":"10.1515\/mgmc-2020-0005","article-title":"Modified Zagreb connection indices of the T-sum graphs","volume":"43","author":"Ali","year":"2020","journal-title":"Main Group Met. Chem."},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Cao, J., Ali, U., Javaid, M., and Huang, C. (2020). Zagreb connection indices of molecular graphs based on operations. Complexity, 2020.","DOI":"10.1155\/2020\/7385682"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"9846913","DOI":"10.1155\/2019\/9846913","article-title":"Zagreb connection indices of subdivision and semi-total point operations on graphs","volume":"2019","author":"Tang","year":"2019","journal-title":"J. Chem."},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Akram, M. (2018). Single-Valued Neutrosophic Graphs, Infosys Science Foundation Series in Mathematical Sciences, Springer.","DOI":"10.1007\/978-981-13-3522-8"},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Akram, M. (2019). m-Polar Fuzzy Graphs, Studies in Fuzziness and Soft Computing, Springer.","DOI":"10.1007\/978-3-030-03751-2"},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Akram, M., and Zafar, F. (2020). Hybrid Soft Computing Models Applied to Graph Theory, Studies in Fuzziness and Soft Computing, Springer.","DOI":"10.1007\/978-3-030-16020-3"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"6019517","DOI":"10.1155\/2021\/6019517","article-title":"Computing analysis for first zagreb connection index and coindex of resultant graphs","volume":"2021","author":"Javaid","year":"2021","journal-title":"Math. Probl. Eng."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"31","DOI":"10.1080\/16583655.2019.1699227","article-title":"Extremal trees for the modified first Zagreb connection index with fixed number of segments or vertices of degree 2","volume":"14","author":"Noureen","year":"2020","journal-title":"J. Taibah Univ. Sci."},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Liu, J.-B., Raza, Z., and Javaid, M. (2020). Zagreb Connection Numbers for Cellular Neural Networks. Discret. Dyn. Nat. Soc., 2020.","DOI":"10.1155\/2020\/8038304"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"4107","DOI":"10.3233\/JIFS-200659","article-title":"On the zagreb connection indices of hex and honeycomb networks","volume":"40","author":"Bataineh","year":"2021","journal-title":"J. Intell. Fuzzy Syst."},{"key":"ref_20","unstructured":"Raza, Z. (2020). Zagreb Connection Indices for Some Benzenoid Systems. Polycycl. Aromat. Compd., 1\u201314."},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Ducoffe, G., Marinescu-Ghemeci, R., Obreja, C., Popa, A., and Tache, R. (2018, January 20\u201323). Extremal graphs with respect to the modified first Zagreb connection index. Proceedings of the 2018 20th International Symposium on Symbolic and Numeric Algorithms for Scientific Computing (SYNASC), Timisoara, Romania.","DOI":"10.1109\/SYNASC.2018.00033"},{"key":"ref_22","first-page":"262","article-title":"The Hosoya Polynomial of the Schreier Graphs of the Group and the Basilica Group","volume":"5","author":"Sahin","year":"2020","journal-title":"Turk. J. Sci."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"6","DOI":"10.47443\/dml.2021.0085","article-title":"Relating graph energy and Sombor index","volume":"8","author":"Ulker","year":"2021","journal-title":"Discret. Math. Lett."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"51","DOI":"10.46793\/match.87-1.051U","article-title":"The energy and Sombor index of graphs","volume":"87","author":"Ulker","year":"2022","journal-title":"MATCH Commun. Math. Comput. Chem."},{"key":"ref_25","first-page":"731","article-title":"Study of some topological invariants of subdivided mk graphs","volume":"2","author":"Afzal","year":"2020","journal-title":"Eurasian Chem. Commun."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"1183","DOI":"10.1080\/09720529.2020.1809112","article-title":"On some degree based topological indices of mk-graph","volume":"23","author":"De","year":"2020","journal-title":"J. Discret. Math. Sci. Cryptogr."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"1279","DOI":"10.1080\/09720529.2020.1819677","article-title":"Study of some connectivity index of subdivided mk graphs of ladder and triangular ladder graph","volume":"23","author":"Cancan","year":"2020","journal-title":"J. Discret. Math. Sci. Cryptogr."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"5539904","DOI":"10.1155\/2021\/5539904","article-title":"Study of Carbon Nanocones via Connection Zagreb Indices","volume":"2021","author":"Asif","year":"2021","journal-title":"Math. Probl. Eng."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"6657039","DOI":"10.1155\/2021\/6657039","article-title":"An Approach to the Extremal Inverse Degree Index for Families of Graphs with Transformation Effect","volume":"2021","author":"Asif","year":"2021","journal-title":"J. Chem."},{"key":"ref_30","doi-asserted-by":"crossref","unstructured":"Asif, M., Kizielewicz, B., Rehman, A.U., Hussain, M., and Sa\u0142abun, W. (2021). Study of \u03b8\u03d5 Networks via Zagreb Connection Indices. Symmetry, 13.","DOI":"10.3390\/sym13111991"},{"key":"ref_31","first-page":"283","article-title":"Topological indices of the mk-graph","volume":"24","author":"Ayache","year":"2017","journal-title":"J. Assoc. Arab. Univ. Basic Appl. Sci."}],"container-title":["Information"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/2.zoppoz.workers.dev:443\/https\/www.mdpi.com\/2078-2489\/13\/4\/179\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T22:47:44Z","timestamp":1760136464000},"score":1,"resource":{"primary":{"URL":"https:\/\/2.zoppoz.workers.dev:443\/https\/www.mdpi.com\/2078-2489\/13\/4\/179"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,3,31]]},"references-count":31,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2022,4]]}},"alternative-id":["info13040179"],"URL":"https:\/\/2.zoppoz.workers.dev:443\/https\/doi.org\/10.3390\/info13040179","relation":{},"ISSN":["2078-2489"],"issn-type":[{"type":"electronic","value":"2078-2489"}],"subject":[],"published":{"date-parts":[[2022,3,31]]}}}