{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,12,20]],"date-time":"2025-12-20T22:08:49Z","timestamp":1766268529190,"version":"build-2065373602"},"reference-count":35,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2016,12,29]],"date-time":"2016-12-29T00:00:00Z","timestamp":1482969600000},"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>A topological index of graph G is a numerical parameter related to G, which characterizes its topology and is preserved under isomorphism of graphs. Properties of the chemical compounds and topological indices are correlated. In this report, we compute closed forms of first Zagreb, second Zagreb, and forgotten polynomials of generalized prism and toroidal polyhex networks. We also compute hyper-Zagreb index, first multiple Zagreb index, second multiple Zagreb index, and forgotten index of these networks. Moreover we gave graphical representation of our results, showing the technical dependence of each topological index and polynomial on the involved structural parameters.<\/jats:p>","DOI":"10.3390\/sym9010005","type":"journal-article","created":{"date-parts":[[2016,12,29]],"date-time":"2016-12-29T10:16:20Z","timestamp":1483006580000},"page":"5","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":15,"title":["Some Algebraic Polynomials and Topological Indices of Generalized Prism and Toroidal Polyhex Networks"],"prefix":"10.3390","volume":"9","author":[{"given":"Muhammad","family":"Ajmal","sequence":"first","affiliation":[{"name":"Department of Mathematics, Government Muhammdan Anglo Orintal College, Lahore 54000, Pakistan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5488-0467","authenticated-orcid":false,"given":"Waqas","family":"Nazeer","sequence":"additional","affiliation":[{"name":"Division of Science and Technology, University of Education, Lahore 54000, Pakistan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Mobeen","family":"Munir","sequence":"additional","affiliation":[{"name":"Division of Science and Technology, University of Education, Lahore 54000, Pakistan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Shin","family":"Kang","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Research Institute of Natural Science, Gyeongsang National University, Jinju 52828, Korea"},{"name":"Center for General Education, China Medical University, Taichung 40402, Taiwan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Young","family":"Kwun","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Dong-A University, Busan 49315, Korea"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2016,12,29]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"788","DOI":"10.1021\/ci9900175","article-title":"On topological indices, boiling points, and cycloalkanes","volume":"39","author":"Rucker","year":"1999","journal-title":"J. Chem. Inf. Comput. Sci."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"1001","DOI":"10.1021\/ci9603689","article-title":"A Comparison of the Schultz molecular topological index with the Wiener index","volume":"36","author":"Gutman","year":"1996","journal-title":"J. Chem. Inf. Comput. Sci."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"336","DOI":"10.1016\/j.cplett.2011.01.033","article-title":"On a class of distance-based molecular structure descriptors","volume":"503","author":"Graovac","year":"2011","journal-title":"Chem. Phys. Lett."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"3017","DOI":"10.1016\/j.camwa.2011.03.089","article-title":"A general modeling of some vertex-degree based topological indices in benzenoid systems and phenylenes","volume":"61","author":"Deng","year":"2011","journal-title":"Comp. Math. Appl."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"147","DOI":"10.1016\/0166-218X(95)00081-2","article-title":"The Clar covering polynomial of hexagonal systems","volume":"69","author":"Zhang","year":"1996","journal-title":"Discret. Appl. Math."},{"key":"ref_6","unstructured":"West, D.B. (1996). An Introduction to Graph Theory, Prentice-Hall."},{"key":"ref_7","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_8","first-page":"137","article-title":"Note on multiple Zagreb indices","volume":"3","author":"Ghorbani","year":"2012","journal-title":"Iran. J. Math. Chem."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"1184","DOI":"10.1007\/s10910-015-0480-z","article-title":"A forgotten topological index","volume":"53","author":"Furtula","year":"2015","journal-title":"J. Math. Chem."},{"key":"ref_10","first-page":"59","article-title":"Sadhana polynomial in nano-dendrimers","volume":"26","author":"Diudea","year":"2010","journal-title":"Carpath. J. Math."},{"key":"ref_11","first-page":"538","article-title":"PI, Szeged and edge Szeged of an infinite family of nanostar dendrimers","volume":"47","author":"Ashrafi","year":"2008","journal-title":"Indian J. Chem."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"462","DOI":"10.1016\/j.amc.2014.05.105","article-title":"Entropy bounds for dendrimers","volume":"242","author":"Chen","year":"2014","journal-title":"Appl. Math. Comput."},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Munir, M., Nazeer, W., Rafique, S., and Kang, S.M. (2016). M-polynomial and related topological indices of Nanostar dendrimers. Symmetry, 8.","DOI":"10.3390\/sym8090097"},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Munir, M., Nazeer, W., Rafique, S., Nizami, A.R., and Kang, S.M. (2016). M-polynomial and degree-based topological indices of titania nanotubes. Symmetry, 8.","DOI":"10.3390\/sym8110117"},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Munir, M., Nazeer, W., Shahzadi, Z., and Kang, S.M. (2016). Some invariants of circulant graphs. Symmetry, 8.","DOI":"10.3390\/sym8110134"},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Munir, M., Nazeer, W., Rafique, S., and Kang, S.M. (2016). M-Polynomial and Degree-Based Topological Indices of Polyhex Nanotubes. Symmetry, 8.","DOI":"10.3390\/sym8120149"},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Ajmal, M., Nazeer, W., Munir, M., Kwun, Y.C., and Kang, S.M. (2016). M-polynomials and topological indices of generalized prism and toroidal polyhex networks. Symmetry, Under review.","DOI":"10.3390\/sym9010005"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"137","DOI":"10.1016\/j.disc.2003.11.009","article-title":"On L(2,1)-Labelling of Cartesian Products of Paths and Cycles","volume":"283","author":"Kuo","year":"2004","journal-title":"Discret. Math."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"2867","DOI":"10.1016\/j.dam.2007.11.019","article-title":"On L(d,1)-Labeling of Cartesian Product of a Path","volume":"156","author":"Chiang","year":"2008","journal-title":"Discret. Appl. Math."},{"key":"ref_20","first-page":"884","article-title":"Computing topological indices of certain networks","volume":"18","author":"Siddiqui","year":"2016","journal-title":"J. Optoelectron. Adv. Mater."},{"key":"ref_21","first-page":"697","article-title":"Incidence Colorings of Cartesian Products of Graphs over Path and Cycles","volume":"40","author":"Deming","year":"2011","journal-title":"Adv. Math."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"247","DOI":"10.1016\/S0166-218X(97)00091-7","article-title":"On domination numbers of Cartesian product of paths","volume":"80","author":"Gravier","year":"1997","journal-title":"Discret. Appl. Math."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"439","DOI":"10.1016\/j.endm.2005.06.062","article-title":"Edge addition number of Cartesian product of paths and cycles","volume":"22","author":"Lai","year":"2005","journal-title":"Electron. Notes Discret. Math."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"479","DOI":"10.1016\/j.dam.2011.11.004","article-title":"Global defensive alliances of trees and Cartesian product of paths and cycles","volume":"160","author":"Chang","year":"2012","journal-title":"Discret. Appl. Math."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"550","DOI":"10.1021\/ci990066h","article-title":"Fullerenes as tilings of surfaces","volume":"40","author":"Deza","year":"2000","journal-title":"J. Chem. Inf. Comput. Sci."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"1256","DOI":"10.1021\/ci980486r","article-title":"How to Enumerate the Connectional Isomers of a Toroidal Polyhex Fullerene","volume":"38","author":"Kirby","year":"1998","journal-title":"J. Chem. Inf. Comput. Sci."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"3868","DOI":"10.1002\/chem.201002758","article-title":"Optical and vibrational properties of toroidal carbon nanotubes","volume":"17","author":"Beuerle","year":"2011","journal-title":"Chem. Eur. J."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"1023","DOI":"10.1137\/080712763","article-title":"On k-resonant fullerene graphs","volume":"23","author":"Ye","year":"2009","journal-title":"SIAM J. Discret. Math."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"2384","DOI":"10.1016\/j.disc.2011.06.018","article-title":"Toroidal fullerenes with the Cayley graph structures","volume":"311","author":"Kang","year":"2011","journal-title":"Discret. Math."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"154","DOI":"10.1016\/j.amc.2014.11.069","article-title":"On topological indices of fullerenes","volume":"251","author":"Baca","year":"2015","journal-title":"Appl. Math. Comput."},{"key":"ref_31","doi-asserted-by":"crossref","unstructured":"Mehranian, Z., and Ashrafi, A.R. (2016). Topological Indices of 3-Generalized Fullerenes, Distance, Symmetry, and Topology in Carbon Nanomaterials, Springer International Publishing.","DOI":"10.1007\/978-3-319-31584-3_17"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"508","DOI":"10.1007\/s10910-006-9211-9","article-title":"GTI-space: The space of generalized topological indices","volume":"43","author":"Ernesto","year":"2008","journal-title":"J. Math. Chem."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"1109","DOI":"10.1021\/ma402131j","article-title":"Unidirectional Alignment of Block Copolymer Films Induced by Expansion of a Permeable Elastomer during Solvent Vapor Annealing","volume":"47","author":"Qiang","year":"2014","journal-title":"Macromolecules"},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"4387","DOI":"10.1021\/ma500934e","article-title":"Supramolecular Multiblock Polystyrene\u2013Polyisobutylene Copolymers via Ionic Interactions","volume":"47","author":"Zhang","year":"2014","journal-title":"Macromolecules"},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"12472","DOI":"10.1021\/acs.langmuir.5b01799","article-title":"Tailor-Made Fluorinated Copolymer\/Clay Nanocomposite by Cationic RAFT Assisted Pickering Miniemulsion Polymerization","volume":"31","author":"Chakrabarty","year":"2015","journal-title":"Langmuir"}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/9\/1\/5\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T19:29:30Z","timestamp":1760210970000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/9\/1\/5"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2016,12,29]]},"references-count":35,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2017,1]]}},"alternative-id":["sym9010005"],"URL":"https:\/\/doi.org\/10.3390\/sym9010005","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2016,12,29]]}}}