{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T01:02:47Z","timestamp":1760230967270,"version":"build-2065373602"},"reference-count":33,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2022,8,17]],"date-time":"2022-08-17T00:00:00Z","timestamp":1660694400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>The A\u03b3 matrix of a graph G is determined by A\u03b3(G)=(1\u2212\u03b3)A(G)+\u03b3D(G), where 0\u2264\u03b3\u22641, A(G) and D(G) are the adjacency and the diagonal matrices of node degrees, respectively. In this case, the A\u03b3 matrix brings together the spectral theories of the adjacency, the Laplacian, and the signless Laplacian matrices, and many more \u03b3 adjacency-type matrices. In this paper, we obtain the A\u03b3 eigenvalues of zero divisor graphs of the integer modulo rings and the von Neumann rings. These results generalize the earlier published spectral theories of the adjacency, the Laplacian and the signless Laplacian matrices of zero divisor graphs.<\/jats:p>","DOI":"10.3390\/sym14081710","type":"journal-article","created":{"date-parts":[[2022,8,17]],"date-time":"2022-08-17T03:15:27Z","timestamp":1660706127000},"page":"1710","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":8,"title":["A\u03b3 Eigenvalues of Zero Divisor Graph of Integer Modulo and Von Neumann Regular Rings"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-1381-0291","authenticated-orcid":false,"given":"Bilal Ahmad","family":"Rather","sequence":"first","affiliation":[{"name":"Mathematical Sciences Department, College of Science, United Arab Emirate University, Al Ain 15551, Abu Dhabi, United Arab Emirates"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7788-791X","authenticated-orcid":false,"given":"Fawad","family":"Ali","sequence":"additional","affiliation":[{"name":"Institute of Numerical Sciences, Kohat University of Science & Technology, Kohat 26000, Khyber Pakhtunkhwa, Pakistan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4361-7786","authenticated-orcid":false,"given":"Asad","family":"Ullah","sequence":"additional","affiliation":[{"name":"Department of Mathematical Sciences, University of Lakki Marwat, Lakki Marwat 28420, Khyber Pakhtunkhwa, Pakistan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1511-9760","authenticated-orcid":false,"given":"Nahid","family":"Fatima","sequence":"additional","affiliation":[{"name":"Department of Math & Sciences, Prince Sultan University, Riyadh 11586, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Rahim","family":"Dad","sequence":"additional","affiliation":[{"name":"Department of Mathematics, University of Science and Technology Bannu, Bannu 28100, Khyber Pakhtunkhwa, Pakistan"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2022,8,17]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Cvetkovi\u0107, D.M., Rowlison, P., and Simi\u0107, S. 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