{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,8]],"date-time":"2026-03-08T06:38:57Z","timestamp":1772951937706,"version":"3.50.1"},"reference-count":12,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2020,4,2]],"date-time":"2020-04-02T00:00:00Z","timestamp":1585785600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100004055","name":"King Fahd University of Petroleum and Minerals","doi-asserted-by":"publisher","award":["IN171003"],"award-info":[{"award-number":["IN171003"]}],"id":[{"id":"10.13039\/501100004055","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>We present lower and upper bounds on the general multiplicative Zagreb indices for bicyclic graphs of a given order and number of pendant vertices. Then, we generalize our methods and obtain bounds for the general multiplicative Zagreb indices of tricyclic graphs, tetracyclic graphs and graphs of given order, size and number of pendant vertices. We show that all our bounds are sharp by presenting extremal graphs including graphs with symmetries. Bounds for the classical multiplicative Zagreb indices are special cases of our results.<\/jats:p>","DOI":"10.3390\/sym12040514","type":"journal-article","created":{"date-parts":[[2020,4,2]],"date-time":"2020-04-02T13:39:34Z","timestamp":1585834774000},"page":"514","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":10,"title":["General Multiplicative Zagreb Indices of Graphs with a Small Number of Cycles"],"prefix":"10.3390","volume":"12","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-3641-290X","authenticated-orcid":false,"given":"Monther R.","family":"Alfuraidan","sequence":"first","affiliation":[{"name":"Department of Mathematics and Statistics, King Fahd University of Petroleum and Minerals, Dhahran 31261, Saudi Arabia"}]},{"given":"Tom\u00e1\u0161","family":"Vetr\u00edk","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Applied Mathematics, University of the Free State, Bloemfontein 9300, South Africa"}]},{"given":"Selvaraj","family":"Balachandran","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Applied Mathematics, University of the Free State, Bloemfontein 9300, South Africa"},{"name":"Department of Mathematics, School of Arts, Sciences and Humanities, SASTRA Deemed University, Thanjavur 613401, India"}]}],"member":"1968","published-online":{"date-parts":[[2020,4,2]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"166","DOI":"10.1016\/j.dam.2017.04.024","article-title":"On extremal multiplicative Zagreb indices of trees with given number of vertices of maximum degree","volume":"227","author":"Wang","year":"2017","journal-title":"Discrete Appl. Math."},{"key":"ref_2","first-page":"241","article-title":"A unified approach to extremal multiplicative Zagreb indices for trees, unicyclic and bicyclic graphs","volume":"68","author":"Xu","year":"2012","journal-title":"MATCH Commun. Math. Comput. Chem."},{"key":"ref_3","unstructured":"Alfuraidan, M.R., Balachandran, S., and Vetr\u00edk, T. General multiplicative Zagreb indices of unicyclic graphs, submitted for publication."},{"key":"ref_4","first-page":"231","article-title":"Sharp upper bounds for multiplicative Zagreb indices","volume":"68","author":"Liu","year":"2012","journal-title":"MATCH Commun. Math. Comput. Chem."},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Wang, S., Wang, C., Chen, L., Liu, J.-B., and Shao, Z. (2018). Maximizing and minimizing multiplicative Zagreb indices of graphs subject to given number of cut edges. Mathematics, 6.","DOI":"10.3390\/math6110227"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"147","DOI":"10.1016\/j.dam.2015.06.028","article-title":"Note on the multiplicative Zagreb indices","volume":"198","author":"Kazemi","year":"2016","journal-title":"Discrete Appl. Math."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"287","DOI":"10.7494\/OpMath.2016.36.3.287","article-title":"Multiplicative Zagreb indices and coindices of some derived graphs","volume":"36","author":"Basavanagoud","year":"2016","journal-title":"Opuscula Math."},{"key":"ref_8","first-page":"417","article-title":"Extremal values of total multiplicative sum Zagreb index and first multiplicative sum Zagreb coindex on unicyclic and bicyclic graphs","volume":"78","author":"Popivoda","year":"2017","journal-title":"MATCH Commun. Math. Comput. Chem."},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Liu, J.-B., Ali, B., Malik, M.A., Siddiqui, H.M.A., and Imran, M. (2019). Reformulated Zagreb indices of some derived graphs. Mathematics, 7.","DOI":"10.3390\/math7040366"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"945","DOI":"10.3390\/math3040945","article-title":"Reformulated first Zagreb index of some graph operations","volume":"3","author":"De","year":"2015","journal-title":"Mathematics"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"341","DOI":"10.1016\/j.dam.2018.03.084","article-title":"General multiplicative Zagreb indices of trees","volume":"247","author":"Balachandran","year":"2018","journal-title":"Discrete Appl. Math."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"1953","DOI":"10.1016\/j.dam.2010.08.005","article-title":"Sharp bounds for the Zagreb indices of bicyclic graphs with k-pendant vertices","volume":"158","author":"Zhao","year":"2010","journal-title":"Discrete Appl. Math."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/12\/4\/514\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T09:14:43Z","timestamp":1760174083000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/12\/4\/514"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,4,2]]},"references-count":12,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2020,4]]}},"alternative-id":["sym12040514"],"URL":"https:\/\/doi.org\/10.3390\/sym12040514","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2020,4,2]]}}}