{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T00:58:44Z","timestamp":1760057924589,"version":"build-2065373602"},"reference-count":23,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2025,2,28]],"date-time":"2025-02-28T00:00:00Z","timestamp":1740700800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Ministry of Tribal Affairs, Govt. of India via MOTA Award","award":["202122-NFST-ARU-00329"],"award-info":[{"award-number":["202122-NFST-ARU-00329"]}]},{"name":"Kholood Alnefaie, Department of Mathematics, College of Science, Taibah University, Madinah 42353, Saudi Arabia","award":["202122-NFST-ARU-00329"],"award-info":[{"award-number":["202122-NFST-ARU-00329"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>In this article, some properties of the zero-divisor graph \u0393(Zn) are investigated when n is a square-free positive integer. It is shown that the zero-divisor graph \u0393(Zn) of ring Zn is a (2k\u22122)-partite graph when the prime decomposition of n contains k distinct square-free primes using the method of congruence relation. We present some examples, accompanied by graphic representations, to achieve the desired results. It is also obtained that the zero-divisor graph \u0393(Zn) is Eulerian if n is a square-free odd integer. Since Zn is a semisimple ring when n is square-free, the results can be generalized to characterize semisimple rings and modules, as well as rings satisfying Artinian and Noetherian conditions through the properties of their zero-divisor graphs. We endeavored to show that \u0393(R) is a partite graph with a certain condition on n and also that \u0393(R) is a complete graph when n=p2 for a prime p as part of a corollary. To prove these results, we employed the assistance of several theoretic congruence relations that grabbed our attention, making the investigation more interesting.<\/jats:p>","DOI":"10.3390\/axioms14030180","type":"journal-article","created":{"date-parts":[[2025,2,28]],"date-time":"2025-02-28T08:05:54Z","timestamp":1740729954000},"page":"180","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["On Zero-Divisor Graphs of Zn When n Is Square-Free"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-2723-446X","authenticated-orcid":false,"given":"Kholood","family":"Alnefaie","sequence":"first","affiliation":[{"name":"Department of Mathematics, College of Science, Taibah University, Madinah 42353, Saudi Arabia"}]},{"given":"Nanggom","family":"Gammi","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Rajiv Gandhi University, Rono Hills, Doimukh 791112, India"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0652-1002","authenticated-orcid":false,"given":"Saifur","family":"Rahman","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Jamia Millia Islamia, New Delhi 110025, India"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5162-7522","authenticated-orcid":false,"given":"Shakir","family":"Ali","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Science, Aligarh Muslim University, Aligarh 202002, India"}]}],"member":"1968","published-online":{"date-parts":[[2025,2,28]]},"reference":[{"key":"ref_1","unstructured":"Aguilar, C.O. (2021). An Introduction to Algebraic Graph Theory, State University of New York."},{"key":"ref_2","unstructured":"Buckley, F., and Lewinter, M. (2003). A Friendly Introduction to Graph Theory, Prentice-Hall."},{"key":"ref_3","unstructured":"Niven, I., Zukerman, H.S., and Montgomery, H.L. (1991). An Introduction to the Theory of Numbers, Wiley India Ltd."},{"key":"ref_4","unstructured":"West, D.B. (2003). Introduction to Graph Theory, Prentice Hall of India Pvt. Ltd."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"462","DOI":"10.1016\/j.jalgebra.2005.07.007","article-title":"Zero-divisor graphs of non-commutative rings","volume":"296","author":"Akbari","year":"2006","journal-title":"J. Algebra"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"3073","DOI":"10.1080\/00927870802110888","article-title":"On the Zero-Divisor Graph of a Ring","volume":"36","author":"Anderson","year":"2008","journal-title":"Commun. Algebra"},{"key":"ref_7","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_8","doi-asserted-by":"crossref","unstructured":"Bondy, J.A., and Murty, U.S.R. (1976). Graph Theory with Applications, Elsevier.","DOI":"10.1007\/978-1-349-03521-2"},{"key":"ref_9","first-page":"93","article-title":"Automorphisms and zero divisor graphs of a commutative rings","volume":"1","author":"Demeyer","year":"2002","journal-title":"Int. J. Commut. Rings"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"407","DOI":"10.4153\/CMB-2011-156-1","article-title":"On Domination in Zero-Divisor Graphs","volume":"56","author":"Rad","year":"2013","journal-title":"Can. Math. Bull."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"755","DOI":"10.1016\/j.disc.2008.01.044","article-title":"On bipartite zero-divisor graphs","volume":"309","author":"Lu","year":"2009","journal-title":"Discret. Math."},{"key":"ref_12","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_13","doi-asserted-by":"crossref","first-page":"17","DOI":"10.1007\/s40065-012-0018-1","article-title":"The subgroup graph of a group","volume":"1","author":"Anderson","year":"2012","journal-title":"Arab. J. Math."},{"key":"ref_14","first-page":"226","article-title":"On zero-divisor graphs of commutative rings without identity","volume":"19","author":"Chelvam","year":"2020","journal-title":"J. Algebra Its Appl."},{"key":"ref_15","first-page":"188","article-title":"A Characterization of Bipartite Zero-Divisor Graphs","volume":"57","author":"Jafari","year":"2014","journal-title":"Can. Math. Soc."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"263","DOI":"10.1007\/s40009-019-00836-8","article-title":"Beck\u2019s Zero-Divisor Graph in the Realm of signed Graph","volume":"3","author":"Sinha","year":"2020","journal-title":"Natl. Acad. Sci. Lett."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"847","DOI":"10.1016\/S0021-8693(03)00435-6","article-title":"When a zero-divisor graph is planar or complete r-partite graph","volume":"274","author":"Akbari","year":"2004","journal-title":"J. Algebra"},{"key":"ref_18","first-page":"203","article-title":"The zero-divisor graph of a non commutative ring","volume":"1","author":"Redmond","year":"2002","journal-title":"Int. J. Commut. Rings"},{"key":"ref_19","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_20","doi-asserted-by":"crossref","first-page":"543","DOI":"10.1016\/j.jpaa.2006.10.007","article-title":"On the diameter and girth of a zero-divisor graph","volume":"210","author":"Anderson","year":"2007","journal-title":"J. Pure Appl. Algebra"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"17","DOI":"10.1021\/ja01193a005","article-title":"Structural determination of paraffin boiling points","volume":"69","author":"Weiner","year":"1947","journal-title":"J. Am. Chem. Soc."},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Dolzan, D. (2023). The Weiner index and the Weiner Complexity of the zero-divisor graph of a ring. arXiv.","DOI":"10.2139\/ssrn.4346337"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"267","DOI":"10.1016\/j.laa.2019.08.015","article-title":"Laplacian eigenvalues of the zero divisor graph of the ring Zn","volume":"584","author":"Chattopadhyay","year":"2020","journal-title":"Linear Algebra Its Appl."}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/14\/3\/180\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,9]],"date-time":"2025-10-09T16:44:39Z","timestamp":1760028279000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/14\/3\/180"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,2,28]]},"references-count":23,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2025,3]]}},"alternative-id":["axioms14030180"],"URL":"https:\/\/doi.org\/10.3390\/axioms14030180","relation":{},"ISSN":["2075-1680"],"issn-type":[{"type":"electronic","value":"2075-1680"}],"subject":[],"published":{"date-parts":[[2025,2,28]]}}}