{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,2]],"date-time":"2026-01-02T07:48:45Z","timestamp":1767340125193,"version":"build-2065373602"},"reference-count":21,"publisher":"MDPI AG","issue":"10","license":[{"start":{"date-parts":[[2023,10,18]],"date-time":"2023-10-18T00:00:00Z","timestamp":1697587200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100004242","name":"Princess Nourah bint Abdulrahman University","doi-asserted-by":"publisher","award":["PNURSP2023R231"],"award-info":[{"award-number":["PNURSP2023R231"]}],"id":[{"id":"10.13039\/501100004242","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>Let R be a commutative ring with identity, and\u00a0Z(R)\u00a0be the set of zero-divisors of R. The weakly zero-divisor graph of R denoted by\u00a0W\u0393(R)\u00a0is an undirected (simple) graph with vertex set\u00a0Z(R)*, and two distinct vertices x and y are adjacent, if and only if there exist\u00a0r\u2208ann(x)\u00a0and\u00a0s\u2208ann(y), such that\u00a0rs=0. Importantly, it is worth noting that\u00a0W\u0393(R)\u00a0contains the zero-divisor graph\u00a0\u0393(R)\u00a0as a subgraph. It is known that graph theory applications play crucial roles in different areas one of which is chemical graph theory that deals with the applications of graph theory to solve molecular problems. Analyzing Zagreb indices in chemical graph theory provides numerical descriptors for molecular structures, aiding in property prediction and drug design. These indices find applications in QSAR modeling and chemical informatics, contributing to efficient compound screening and optimization. They are essential tools for advancing pharmaceutical and material science research. This research article focuses on the basic properties of the weakly zero-divisor graph of the ring\u00a0Zp\u00d7Zt\u00d7Zs, denoted by\u00a0W\u0393(Zp\u00d7Zt\u00d7Zs), where p, t, and s are prime numbers that may not necessarily be distinct and greater than 2. Moreover, this study includes the examination of various indices and coindices such as the first and second Zagreb indices and coindices, as well as the first and second multiplicative Zagreb indices and coindices of\u00a0W\u0393(Zp\u00d7Zt\u00d7Zs).<\/jats:p>","DOI":"10.3390\/axioms12100987","type":"journal-article","created":{"date-parts":[[2023,10,18]],"date-time":"2023-10-18T10:36:56Z","timestamp":1697625416000},"page":"987","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Analysis of the Zagreb Indices over the Weakly Zero-Divisor Graph of the Ring\u00a0Zp\u00d7Zt\u00d7Zs"],"prefix":"10.3390","volume":"12","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-3955-7941","authenticated-orcid":false,"given":"Nadeem ur","family":"Rehman","sequence":"first","affiliation":[{"name":"Department of Mathematics, Aligarh Muslim University, Aligarh 202002, India"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7856-2861","authenticated-orcid":false,"given":"Amal S.","family":"Alali","sequence":"additional","affiliation":[{"name":"Department of Mathematical Sciences, College of Science, Princess Nourah bint Abdulrahman University, P.O. Box 84428, Riyadh 11671, Saudi Arabia"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4743-6859","authenticated-orcid":false,"given":"Shabir Ahmad","family":"Mir","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Aligarh Muslim University, Aligarh 202002, India"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8817-4336","authenticated-orcid":false,"given":"Mohd","family":"Nazim","sequence":"additional","affiliation":[{"name":"School of Basic and Applied Sciences, Faculty of Science and Technology, JSPM University, Pune 412207, India"}]}],"member":"1968","published-online":{"date-parts":[[2023,10,18]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"208","DOI":"10.1016\/0021-8693(88)90202-5","article-title":"Coloring of commutative rings","volume":"116","author":"Beck","year":"1988","journal-title":"J. Algebra"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"434","DOI":"10.1006\/jabr.1998.7840","article-title":"The zero-divisor graph of a commutative ring","volume":"217","author":"Anderson","year":"1999","journal-title":"J. Algebra"},{"key":"ref_3","first-page":"390","article-title":"Analyzing special parameters over zero-divisor graphs","volume":"1479","year":"2012","journal-title":"AIP Conf. Proc."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"500","DOI":"10.1006\/jabr.1993.1171","article-title":"Beck\u2019s coloring of a commutative ring","volume":"159","author":"Anderson","year":"1993","journal-title":"J. Algebra"},{"key":"ref_5","first-page":"144","article-title":"Analysis of graph parameters associated with zero-divisor graphs of commutative rings","volume":"6","year":"2018","journal-title":"New Trends Math. Sci."},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Gross, J.L., and Yellen, J. (2004). Handbook of Graph Theory, CRC Press.","DOI":"10.1201\/9780203490204"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"375","DOI":"10.1134\/S1995080215040071","article-title":"On finite rings in which zero-divisor graphs satisfy the Dirac\u2019s condition","volume":"36","author":"Kuzmina","year":"2015","journal-title":"Lobachevskii J. Math"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"41","DOI":"10.3103\/S1066369X14120056","article-title":"Finite rings with some restrictions on zero-divisor graphs","volume":"58","author":"Kuzmina","year":"2014","journal-title":"Russian Math."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"1250055","DOI":"10.1142\/S0219498812500557","article-title":"Finite rings with Eulerian zero-divisor graphs","volume":"11","author":"Kuzmina","year":"2012","journal-title":"J. Algebra Appli."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"175","DOI":"10.1016\/j.aml.2011.08.010","article-title":"A note on diameter and the degree sequence of a graph","volume":"25","author":"Mukwembi","year":"2012","journal-title":"Appl. Math. Lett."},{"key":"ref_11","first-page":"153","article-title":"On the first Zagreb index and multiplicative Zagreb coindices of graphs","volume":"24","author":"Das","year":"2016","journal-title":"Analele Stiint. Univ. Ovidius Constanta Ser. Mat."},{"key":"ref_12","first-page":"38","article-title":"Analysis of adjacency matrix and neighborhood associated with zero divisor graph of finite commutative rings","volume":"14","author":"Sharma","year":"2011","journal-title":"Int. J. Comput. Appl."},{"key":"ref_13","first-page":"309","article-title":"Some graph theoretical properties over zero-divisor graphs of special finite commutative rings","volume":"22","author":"Togan","year":"2012","journal-title":"Adv. Stud. Contemp. Math."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"105","DOI":"10.33044\/revuma.1677","article-title":"The weakly zero-divisor graph of a commutative ring","volume":"62","author":"Nikmehr","year":"2021","journal-title":"Rev. Union Math. Argent."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"74","DOI":"10.1016\/j.akcej.2018.05.002","article-title":"Beyond the Zagreb indices","volume":"17","author":"Gutman","year":"2018","journal-title":"AKCE Int. J. Graphs Comb."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"645","DOI":"10.3329\/jsr.v12i4.45389","article-title":"Topological Indices of Dendrimers used in Drug Delivery","volume":"12","author":"Jude","year":"2020","journal-title":"J. Sci. Res."},{"key":"ref_17","unstructured":"Pirzada, S. (2012). An Introduction to Graph Theory, Orient Blakswan."},{"key":"ref_18","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_19","doi-asserted-by":"crossref","first-page":"66","DOI":"10.26493\/1855-3974.15.895","article-title":"Vertex weighted Wiener polynomials for composite graphs","volume":"1","year":"2008","journal-title":"Ars. Math. Contemp."},{"key":"ref_20","first-page":"359","article-title":"New local vertex invariants and molecular descriptors based on functions of the vertex degrees","volume":"64","author":"Todeschini","year":"2010","journal-title":"MATCH Commun. Math. Comput. Chem."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"191","DOI":"10.7494\/OpMath.2013.33.1.191","article-title":"On the multiplicative Zagreb coindex of graphs","volume":"33","author":"Xu","year":"2013","journal-title":"Opuscula Math."}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/12\/10\/987\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T21:09:11Z","timestamp":1760130551000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/12\/10\/987"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,10,18]]},"references-count":21,"journal-issue":{"issue":"10","published-online":{"date-parts":[[2023,10]]}},"alternative-id":["axioms12100987"],"URL":"https:\/\/doi.org\/10.3390\/axioms12100987","relation":{},"ISSN":["2075-1680"],"issn-type":[{"type":"electronic","value":"2075-1680"}],"subject":[],"published":{"date-parts":[[2023,10,18]]}}}