{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,27]],"date-time":"2026-05-27T10:02:47Z","timestamp":1779876167133,"version":"3.53.1"},"reference-count":14,"publisher":"F1000 Research Ltd","license":[{"start":{"date-parts":[[2026,1,9]],"date-time":"2026-01-09T00:00:00Z","timestamp":1767916800000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Fallujah Multidisciplinary Science and Innovation"}],"content-domain":{"domain":["f1000research.com"],"crossmark-restriction":false},"short-container-title":["F1000Res"],"abstract":"<ns3:p>Background The study of algebraic structures through graph-theoretic representations provides a powerful visual and combinatorial framework for analyzing ring-theoretic properties. The ideal-based non-zero divisor graph \u2205 I ( Z n ) , constructed from the ring of integers modulo n with respect to a proper ideal I . This graph extends the classical zero-divisor graph framework and serves as a visual and structural invariant for analyzing ideal interactions in finite commutative rings. Methods Using combinatorial graph theory and modular arithmetic, we analyze fundamental properties of \u2205 I ( Z n ) . Vertex degrees, connectivity, and cut-sets are characterized using divisibility conditions and the Euler totient function \u03d5 ( n ) . The analysis distinguishes cases based on the parity and primality of n , as well as the generator of I . Topological indices, including the Zagreb and Randi\u0107 indices, are formulated to quantify structural complexity. Results We establish necessary and sufficient conditions for the connectivity of \u2205 I ( Z n ) , proving it is connected for all n \u2265 10 and any non-zero proper ideal I . For prime n \u2209 { 2 , 3 } , the graph is shown to be complete. General formulas are provided for calculating vertex degrees based on gcd ( x , d ) where I = &lt; d &gt; . Furthermore, the structure and computation cut-sets are characterized for Z p 2 and composite n = xy . Moreover, the domination number \u03b3 ( \u2205 I ( Z n ) )=1 and girth gr ( \u2205 I ( Z n ) )=3 is established for n \u2265 10 . General expressions for Zagreb and Randi\u0107 indices are derived, directly linking graph invariants to n and d . Conclusions The graph \u2205 I ( Z n ) serves as an effective combinatorial invariant for studying the interplay between ideals and zero-divisor structure in Z n . These results establish systematic connections between ring-theoretic properties and graph parameters, enabling both qualitative and quantitative analysis through connectivity, degree distributions, cut-sets, and topological indices.<\/ns3:p>","DOI":"10.12688\/f1000research.172788.1","type":"journal-article","created":{"date-parts":[[2026,1,9]],"date-time":"2026-01-09T10:19:09Z","timestamp":1767953949000},"page":"44","update-policy":"https:\/\/doi.org\/10.12688\/f1000research.crossmark-policy","source":"Crossref","is-referenced-by-count":0,"title":["A Study on the Structure of Ideal-Based Non-Zero Divisor Graphs Associated with Zn"],"prefix":"10.12688","volume":"15","author":[{"given":"Sameer","family":"Kadem","sequence":"first","affiliation":[{"name":"Department of Mathematics Baghdad Baghdad Governorate University of Baghdad"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4764-2946","authenticated-orcid":false,"given":"Ali","family":"Abd Aubad","sequence":"additional","affiliation":[{"name":"Department of Mathematics Baghdad Baghdad Governorate University of Baghdad"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"2560","published-online":{"date-parts":[[2026,1,9]]},"reference":[{"key":"ref1","doi-asserted-by":"publisher","first-page":"23-39","DOI":"10.1007\/978-3-319-51718-6_2","author":"D Anderson","year":"2017","journal-title":"Groups, Modules, and Model Theory - Surveys and Recent Developments."},{"key":"ref2","doi-asserted-by":"publisher","first-page":"23-45","DOI":"10.1007\/978-1-4419-6990-3_2","author":"D Anderson","year":"2011","journal-title":"Commutative Algebra: Noetherian and Non-Noetherian Perspectives."},{"key":"ref3","doi-asserted-by":"publisher","first-page":"500-514","DOI":"10.1006\/jabr.1993.1171","article-title":"Beck\u2019s coloring of a commutative ring.","volume":"159","author":"D Anderson","year":"Aug. 1993","journal-title":"J. Algebra."},{"key":"ref4","article-title":"Structure in Zero-Divisor Graphs of Commutative Rings.","author":"P Livingston","year":"1997"},{"key":"ref5","article-title":"Generalizations and Variations of the Zero-Divisor Graph.","author":"G Mcclurkin"},{"key":"ref6","article-title":"Generalizations of the Zero-Divisor Graph of a Ring.","author":"D Dissertations","year":"2001"},{"key":"ref7","doi-asserted-by":"publisher","first-page":"277-282","DOI":"10.21123\/bsj.2023.8408","article-title":"Quotient Energy of Zero Divisor Graphs and Identity Graphs.","volume":"20","author":"M Kumari","year":"2021","journal-title":"Baghdad Sci. J."},{"key":"ref8","doi-asserted-by":"publisher","first-page":"4425-4443","DOI":"10.1081\/AGB-120022801","article-title":"An ideal-based zero-divisor graph of a commutative ring.","volume":"31","author":"S Redmond","year":"Sep. 2003","journal-title":"Commun. Algebra."},{"key":"ref9","first-page":"975-983","article-title":"The non-zero divisor graph of a ring.","volume":"43","author":"S Kadem","year":"2020","journal-title":"Ital. J. Pure Appl. Math. N."},{"key":"ref10","doi-asserted-by":"publisher","first-page":"2902-2908","DOI":"10.24996\/ijs.2025.66.7.19","article-title":"The Non-Zero Divisor Graph of Prime Ring.","author":"A Majeed","year":"Jul. 2025","journal-title":"Iraqi J. Sci."},{"key":"ref11","doi-asserted-by":"publisher","first-page":"1301-1307","DOI":"10.47974\/JDMSC-2268","article-title":"Some results on the non-zero divisor graphs of Zn.","volume":"28","author":"F Sadiq","year":"Jun. 2025","journal-title":"J. Discret. Math. Sci. Cryptogr."},{"key":"ref12","article-title":"GRAPH THEORY AND app.","author":"J Gross","year":"2019"},{"key":"ref14","article-title":"Cut-sets and Cut-vertices in the Zero-Divisor Graph of \u220fZni.","author":"B Cot\u00e9","year":"2010"},{"key":"ref15","doi-asserted-by":"publisher","first-page":"93-100","DOI":"10.2298\/FIL1201093G","article-title":"A new version of Zagreb indices.","volume":"26","author":"M Ghorbani","year":"2012","journal-title":"Filomat."}],"container-title":["F1000Research"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/f1000research.com\/articles\/15-44\/v1\/xml","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/f1000research.com\/articles\/15-44\/v1\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/f1000research.com\/articles\/15-44\/v1\/iparadigms","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,5,27]],"date-time":"2026-05-27T09:09:56Z","timestamp":1779872996000},"score":1,"resource":{"primary":{"URL":"https:\/\/f1000research.com\/articles\/15-44\/v1"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,1,9]]},"references-count":14,"URL":"https:\/\/doi.org\/10.12688\/f1000research.172788.1","relation":{"has-review":[{"id-type":"doi","id":"10.5256\/f1000research.190544.r449782","asserted-by":"subject"},{"id-type":"doi","id":"10.5256\/f1000research.190544.r449777","asserted-by":"subject"},{"id-type":"doi","id":"10.5256\/f1000research.190544.r449782","asserted-by":"object"},{"id-type":"doi","id":"10.5256\/f1000research.190544.r449777","asserted-by":"object"}]},"ISSN":["2046-1402"],"issn-type":[{"value":"2046-1402","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,1,9]]},"assertion":[{"value":"Indexed","URL":"https:\/\/f1000research.com\/articles\/15-44\/v1#article-reports","order":0,"name":"referee-status","label":"Referee status","group":{"name":"current-referee-status","label":"Current Referee Status"}},{"value":"10.5256\/f1000research.190544.r449782, Nabeel Ezzulddin Arif, Amir A. Majeed, Mathematical Sciences, Tikrit University, Tikrit, Saladin, 27 Jan 2026, version 1, indexed","URL":"https:\/\/f1000research.com\/articles\/15-44\/v1#referee-response-449782","order":0,"name":"referee-response-449782","label":"Referee Report","group":{"name":"article-reports","label":"Article Reports"}},{"value":"10.5256\/f1000research.190544.r449777, Faisal Al-Sharqi, University of Anbar, Ramadi, 10 Feb 2026, version 1, indexed","URL":"https:\/\/f1000research.com\/articles\/15-44\/v1#referee-response-449777","order":1,"name":"referee-response-449777","label":"Referee Report","group":{"name":"article-reports","label":"Article Reports"}},{"value":"<b>Ali Aubad<\/b>; \n<i>Posted: 20 Feb 2026<\/i>; Dear Reviewer,  We are grateful for your thorough and encouraging review of our manuscript. Thank you for recognising this study as a comprehensive and significant work titled &quot;Ideal-Based Non-Zero Divisor Graphs Associated with Zn.&quot; We are pleased that you recognised the significance of the connections we established between graph theory and ring theory.  We sincerely value your insightful comments on the technical quality and presentation of the work. It is reassuring to see the soundness of our reasoning and the clarity of the research's organization, which facilitates the reader's understanding of the results.  We are delighted that you have accepted the manuscript and considered it a valuable contribution to the field of algebraic graph theory.  Thank you again for your time and expertise.  Sincerely,  The Authors","URL":"https:\/\/f1000research.com\/articles\/15-44\/v1#referee-comment-15499","order":2,"name":"referee-comment-15499","label":"Referee Comment","group":{"name":"article-reports","label":"Article Reports"}},{"value":"This research was partially supported by the 4th Annual International Conference on Information and Sciences (AICIS\u201925), organized by the University of Fallujah.\r\nFurther details about the conference can be found at: https:\/\/edas.info\/index.php?c=34198\r\nNo additional grants or financial support were received from public, commercial, or not-for-profit funding agencies for the research, authorship, or publication of this article. The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.","order":3,"name":"grant-information","label":"Grant Information"},{"value":"This is an open access article distributed under the terms of the Creative Commons Attribution Licence, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.","order":0,"name":"copyright-info","label":"Copyright"}]}}