{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,25]],"date-time":"2026-03-25T19:26:55Z","timestamp":1774466815491,"version":"3.50.1"},"reference-count":10,"publisher":"MDPI AG","issue":"10","license":[{"start":{"date-parts":[[2023,10,11]],"date-time":"2023-10-11T00:00:00Z","timestamp":1696982400000},"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 1\u22600 and let Z(R)\u2032 be the set of all non-unit and non-zero elements of ring R. \u0393\u2032(R) denotes the cozero-divisor graph of R and is an undirected graph with vertex set Z(R)\u2032, w\u2209zR, and z\u2209wR if and only if two distinct vertices w and z are adjacent, where qR is the ideal generated by the element q in R. In this article, we investigate the signless Laplacian eigenvalues of the graphs \u0393\u2032(Zn). We also show that the cozero-divisor graph \u0393\u2032(Zp1p2) is a signless Laplacian integral.<\/jats:p>","DOI":"10.3390\/axioms12100957","type":"journal-article","created":{"date-parts":[[2023,10,11]],"date-time":"2023-10-11T02:20:52Z","timestamp":1696990852000},"page":"957","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":["Spectrum of the Cozero-Divisor Graph Associated to Ring Zn"],"prefix":"10.3390","volume":"12","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-3724-7608","authenticated-orcid":false,"given":"Mohd","family":"Rashid","sequence":"first","affiliation":[{"name":"Department of Mathematics, Faculty of Science, 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\/0000-0001-7862-7071","authenticated-orcid":false,"given":"Wasim","family":"Ahmed","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Science, Aligarh Muslim University, Aligarh 202002, India"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0350-9377","authenticated-orcid":false,"given":"Muzibur Rahman","family":"Mozumder","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Science, Aligarh Muslim University, Aligarh 202002, India"}]}],"member":"1968","published-online":{"date-parts":[[2023,10,11]]},"reference":[{"key":"ref_1","first-page":"753","article-title":"The cozero-divisor graph of a commutative ring","volume":"35","author":"Afkhami","year":"2011","journal-title":"Southeast Asian Bull. 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Mathematics, 9.","DOI":"10.3390\/math9050482"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"753","DOI":"10.2140\/involve.2015.8.753","article-title":"Adjacency matrices of zero divisor graphs of integer modulo n","volume":"8","author":"Young","year":"2015","journal-title":"Involve"},{"key":"ref_7","first-page":"935","article-title":"On the cozero-divisor graphs of commutative rings and their complements","volume":"35","author":"Afkhami","year":"2012","journal-title":"Bull. Malays. Math. Sci. Soc."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"1350113","DOI":"10.1142\/S0219498813501132","article-title":"Some results on cozero-divisor graph of a commutative ring","volume":"13","author":"Akbari","year":"2014","journal-title":"J. Algebra Appl."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"1450046","DOI":"10.1142\/S1793830914500463","article-title":"Signless Laplacian and normalized Laplacian on the H-join operation of graphs","volume":"6","author":"Wu","year":"2014","journal-title":"Discrete Math. Algorithms Appl."},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Joita, D.M., and J\u00e4ntschi, L. (2017). Extending the characteristic Polynomial for characterization of C20 Fullerene Congeners. Mathematics, 5.","DOI":"10.3390\/math5040084"}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/12\/10\/957\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T21:04:40Z","timestamp":1760130280000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/12\/10\/957"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,10,11]]},"references-count":10,"journal-issue":{"issue":"10","published-online":{"date-parts":[[2023,10]]}},"alternative-id":["axioms12100957"],"URL":"https:\/\/doi.org\/10.3390\/axioms12100957","relation":{},"ISSN":["2075-1680"],"issn-type":[{"value":"2075-1680","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,10,11]]}}}