{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,6]],"date-time":"2025-11-06T12:34:57Z","timestamp":1762432497877,"version":"build-2065373602"},"reference-count":16,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2025,1,4]],"date-time":"2025-01-04T00:00:00Z","timestamp":1735948800000},"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>The zero-divisor graph of a commutative ring R with a nonzero identity, denoted by \u0393(R), is an undirected graph where the vertex set Z(R)* consists of all nonzero zero-divisors of R. Two distinct vertices a and b in \u0393(R) are adjacent if and only if ab=0. The normalized Laplacian spectrum of zero-divisor graphs has been studied extensively due to its algebraic and combinatorial significance. Notably, Pirzada and his co-authors computed the normalized Laplacian spectrum of \u0393(Zn) for specific values of n in the set {pq,p2q,p3,p4}, where p and q are distinct primes satisfying p&lt;q. Motivated by their work, this article investigates the normalized Laplacian spectrum of \u0393(Zn) for a more general class of n, where n is represented as p1T1p2T2, with p1 and p2 being distinct primes (p1&lt;p2), and T1,T2 are positive integers.<\/jats:p>","DOI":"10.3390\/axioms14010037","type":"journal-article","created":{"date-parts":[[2025,1,6]],"date-time":"2025-01-06T08:08:52Z","timestamp":1736150932000},"page":"37","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["On the Normalized Laplacian Spectrum of the Zero-Divisor Graph of the Commutative Ring Zp1T1p2T2"],"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\/0000-0002-9169-8488","authenticated-orcid":false,"family":"Nazim","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Aligarh Muslim University, Aligarh 202002, India"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3955-7941","authenticated-orcid":false,"given":"Nadeem Ur","family":"Rehman","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Aligarh Muslim University, Aligarh 202002, India"}]}],"member":"1968","published-online":{"date-parts":[[2025,1,4]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"208","DOI":"10.1016\/0021-8693(88)90202-5","article-title":"Coloring of a commutative ring","volume":"116","author":"Beck","year":"1988","journal-title":"J. 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