{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T00:58:08Z","timestamp":1760057888898,"version":"build-2065373602"},"reference-count":29,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2025,2,26]],"date-time":"2025-02-26T00:00:00Z","timestamp":1740528000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"King Khalid University","award":["RGP2\/339\/45"],"award-info":[{"award-number":["RGP2\/339\/45"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>Rc represents commutative rings that have unity elements. The collection of all zero-divisor elements in Rc are represented by D(Rc). We denote an extended zero-divisor graph by the notation \u2138\u2032(Rc) of Rc. This graph has a set of vertices in D(Rc)*. The graph \u2138\u2032(Rc) illustrates interactions among the zero-divisor elements of Rc. Specifically, two different vertices u and y are connected in \u2138\u2032(Rc) iff uRc\u2229Ann(y) is non-null or yRc\u2229Ann(u) is non-null. The main idea for this work is to systematically analyze the ring Rc which is finite for the unique aspect of their extended zero-divisor graph. This study particularly focuses on instances where the extended zero-divisor graph has a genus or crosscap of two. Furthermore, this work aims to thoroughly characterize finite ring Rc wherein the extended zero-divisor graph \u2138\u2032(Rc) has an outerplanarity index of two. Finally, we determine the book thickness of \u2138\u2032(Rc) for genus at most one.<\/jats:p>","DOI":"10.3390\/axioms14030170","type":"journal-article","created":{"date-parts":[[2025,2,26]],"date-time":"2025-02-26T08:28:31Z","timestamp":1740558511000},"page":"170","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Exploring the Embedding of the Extended Zero-Divisor Graph of Commutative Rings"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-4167-3119","authenticated-orcid":false,"given":"Ali","family":"Al Khabyah","sequence":"first","affiliation":[{"name":"Department of Mathematics, College of Science, King Khalid University, Abha 61413, Saudi Arabia"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1175-9704","authenticated-orcid":false,"given":"Moin A.","family":"Ansari","sequence":"additional","affiliation":[{"name":"Department of Mathematics, College of Science, Jazan University, P.O. Box 114, Jazan 45142, Saudi Arabia"}]}],"member":"1968","published-online":{"date-parts":[[2025,2,26]]},"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","doi-asserted-by":"crossref","first-page":"4384683","DOI":"10.1155\/2021\/4384683","article-title":"Classification of rings with toroidal and projective coannihilator graph","volume":"2021","author":"Alanazi","year":"2021","journal-title":"J. Math."},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Anderson, D.F., Asir, T., Badawi, A., and Tamizh, C.T. (2021). 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