{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,12,17]],"date-time":"2025-12-17T12:55:29Z","timestamp":1765976129010,"version":"build-2065373602"},"reference-count":38,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2017,1,23]],"date-time":"2017-01-23T00:00:00Z","timestamp":1485129600000},"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>In this report, we compute closed forms of M-polynomial, first and second Zagreb polynomials and forgotten polynomial for Jahangir graphs Jn,m for all values of m and n. From the M-polynomial, we recover many degree-based topological indices such as first and second Zagreb indices, modified Zagreb index, Symmetric division index, etc. We also compute harmonic index, first and second multiple Zagreb indices and forgotten index of Jahangir graphs. Our results are extensions of many existing results.<\/jats:p>","DOI":"10.3390\/sym9010017","type":"journal-article","created":{"date-parts":[[2017,1,23]],"date-time":"2017-01-23T10:40:33Z","timestamp":1485168033000},"page":"17","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":11,"title":["Some Invariants of Jahangir Graphs"],"prefix":"10.3390","volume":"9","author":[{"given":"Mobeen","family":"Munir","sequence":"first","affiliation":[{"name":"Division of Science and Technology, University of Education, 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":"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":"Muhammad","family":"Qureshi","sequence":"additional","affiliation":[{"name":"COMSATS Institute of Information Technology, Vehari Campus, Vehari, 61100 Pakistan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Abdul","family":"Nizami","sequence":"additional","affiliation":[{"name":"Division of Science and Technology, University of Education, Lahore 54000, Pakistan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Youl","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":[[2017,1,23]]},"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","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":"Klavzar","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":"Comput. 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 I","volume":"69","author":"Zhang","year":"1996","journal-title":"Discret. Appl. Math."},{"key":"ref_6","first-page":"13","article-title":"Some properties of the Wiener polynomials","volume":"125","author":"Gutman","year":"1993","journal-title":"Graph Theory Notes N. Y."},{"key":"ref_7","first-page":"93","article-title":"M-Polynomial, and degree-based topological indices","volume":"6","author":"Deutsch","year":"2015","journal-title":"Iran. J. Math. Chem."},{"key":"ref_8","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_9","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_10","doi-asserted-by":"crossref","unstructured":"Munir, M., Nazeer, W., Rafique, S., Nizami, A.R., and Kang, S.M. (2016). Some Computational Aspects of Triangular Boron Nanotubes. Symmetry, 9.","DOI":"10.20944\/preprints201611.0041.v1"},{"key":"ref_11","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_12","first-page":"42","article-title":"On pebbling Jahangir graph","volume":"5","author":"Lourdusamy","year":"2011","journal-title":"Gen. Math. Notes"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"1193","DOI":"10.12988\/ijcms.2007.07122","article-title":"Domination in Jahangir Graph J2,m","volume":"2","author":"Mojdeh","year":"2007","journal-title":"Int. J. Contemp. Math. Sci."},{"key":"ref_14","unstructured":"Ali, K., Baskoro, E.T., and Tomescu, I. (2007, January 4\u20137). On the Ramsey number of Paths and Jahangir, graph J3,m. Proceedings of the 3rd International Conference on 21st Century Mathematics, Lahore, Pakistan."},{"key":"ref_15","first-page":"221","article-title":"Hosoya Polynomial and Wiener Index of Jahangir graphs J2,m","volume":"7","author":"Farahani","year":"2015","journal-title":"Pac. J. Appl. Math."},{"key":"ref_16","first-page":"91","article-title":"The Wiener Index and Hosoya polynomial of a class of Jahangir graphs J3,m","volume":"3","author":"Farahani","year":"2015","journal-title":"Fundam. J. Math. Math. Sci."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"138","DOI":"10.11648\/j.acm.20160503.17","article-title":"The Wiener Index and the Hosoya Polynomial of the Jahangir Graphs","volume":"5","author":"Wang","year":"2016","journal-title":"Appl. Comput. Math."},{"key":"ref_18","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_19","doi-asserted-by":"crossref","first-page":"211","DOI":"10.1023\/A:1010767517079","article-title":"Wiener index of trees: Theory and applications","volume":"66","author":"Dobrynin","year":"2001","journal-title":"Acta Appl. Math."},{"key":"ref_20","unstructured":"Gutman, I., and Polansky, O.E. (2012). Mathematical Concepts in Organic Chemistry, Springer Science & Business Media."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"6609","DOI":"10.1021\/ja00856a001","article-title":"Characterization of molecular branching","volume":"97","author":"Randic","year":"1975","journal-title":"J. Am. Chem. Soc."},{"key":"ref_22","first-page":"225","article-title":"Graphs of extremal weights","volume":"50","year":"1998","journal-title":"Ars Comb."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"819","DOI":"10.1021\/ci980039b","article-title":"The vertex-connectivity index revisited","volume":"38","author":"Amic","year":"1998","journal-title":"J. Chem. Inf. Comput. Sci."},{"key":"ref_24","first-page":"425","article-title":"On molecular graphs with smallest and greatest zeroth-order general Randic index","volume":"54","author":"Hu","year":"2005","journal-title":"MATCH Commun. Math. Comput. Chem."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"85","DOI":"10.1016\/S0097-8485(02)00016-5","article-title":"Graphs with maximum connectivity index","volume":"27","author":"Caporossi","year":"2003","journal-title":"Comput. Biol. Chem."},{"key":"ref_26","unstructured":"Li, X., and Gutman, I. (2006). Mathematical Aspects of Randic-Type Molecular Descriptors, University of Kragujevac and Faculty of Science Kragujevac."},{"key":"ref_27","unstructured":"Kier, L.B., and Hall, L.H. (1976). Molecular Connectivity in Chemistry and Drug Research, Academic Press."},{"key":"ref_28","unstructured":"Kier, L.B., and Hall, L.H. (1986). Molecular Connectivity in Structure-Activity Analysis, Wiley."},{"key":"ref_29","first-page":"5","article-title":"On history of the Randi\u0107 index and emerging hostility toward chemical graph theory","volume":"59","year":"2008","journal-title":"MATCH Commun. Math. Comput. Chem."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"19","DOI":"10.1016\/S1093-3263(01)00098-5","article-title":"The connectivity index 25 years after","volume":"20","year":"2001","journal-title":"J. Mol. Graph. Model."},{"key":"ref_31","first-page":"113","article-title":"The Zagreb indices 30 years after","volume":"76","year":"2003","journal-title":"Croat. Chem. Acta"},{"key":"ref_32","first-page":"83","article-title":"The first Zagreb index 30 years after","volume":"50","author":"Gutman","year":"2004","journal-title":"MATCH Commun. Math. Comput. Chem."},{"key":"ref_33","first-page":"103","article-title":"Some properties of the second Zagreb index","volume":"52","author":"Das","year":"2004","journal-title":"MATCH Commun. Math. Comput. Chem."},{"key":"ref_34","first-page":"577","article-title":"On Zagreb indices","volume":"59","author":"Trinajstic","year":"2010","journal-title":"Kem. Ind."},{"key":"ref_35","first-page":"483","article-title":"Augmented Zagreb Index of Connected Graphs","volume":"67","author":"Huang","year":"2012","journal-title":"MATCH Commun. Math. Comput. Chem."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"370","DOI":"10.1007\/s10910-010-9677-3","article-title":"Augmented Zagreb index","volume":"48","author":"Furtula","year":"2010","journal-title":"J. Math. Chem."},{"key":"ref_37","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_38","first-page":"137","article-title":"Note on multiple Zagreb indices","volume":"3","author":"Ghorbani","year":"2012","journal-title":"Iran. J. Math. Chem."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/9\/1\/17\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T18:26:49Z","timestamp":1760207209000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/9\/1\/17"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2017,1,23]]},"references-count":38,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2017,1]]}},"alternative-id":["sym9010017"],"URL":"https:\/\/doi.org\/10.3390\/sym9010017","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2017,1,23]]}}}