# Line graphs of unit graphs associated with the direct product of rings

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[1] M. Afkhami and F. Khosh-Ahang, Unit graphs of rings of polynomials and power series, Arab. J. Math. 2 (3) (2013), 233–246. Google Scholar

[2] N. Ashrafi, H. R. Maimani, M.R.Pournaki and S. Yassemi, Unit graphs associated to rings, Comm. Algabra 38 (8) (2010), 2851–2871. Google Scholar

[3] Z. Barati, Line zero divisor graphs, J. Algebra Appl. 20 (9) (2021) 2150154 (13 pages). Google Scholar

[4] M. Imran Bhat and S. Pirzada, On strong metric dimension of zero divisor graphs of rings, The Korean J. Mathematics 27 (3) (2019), 563–580. Google Scholar

[5] Bilal A. Rather, S. Pirzada, T. A. Naikoo and Y. Shang, On Laplacian eigenvalues of the zer divisor graph associated to the ring of integers modulo n, Mathematics 9 (5) (2021), 482. Google Scholar

[6] L. W. Beineke, Characterizations of derived graphs, J. Comb. Theory 9 (1970), 129–135. Google Scholar

[7] R. P. Grimaldi, Graphs from rings, Proceedings of the 20th Southeastern Conference on Com- binatorics, Graph Theory, and Computing (Boca Raton, FL, 1989). Congr. Numer. 71 (1990), 95–103. Google Scholar

[8] F. Heydari and M. J. Nikmehr, The unit graph of a left Artinian ring, Acta Math. Hungar. 139 (1-2) (2013), 134–146. Google Scholar

[9] I. Kaplansky, Commutative Rings, Chicago, Ill.-London, The University of Chicago Press (1974). Google Scholar

[10] S. Kiani, H. R. Maimani, M. R. Pournaki and S. Yassemi, Classification of rings with unit graphs having domination number less than four, Rend. Sem. Mat. Univ. Padova 133 (2015), 173–195. Google Scholar

[11] H. R. Maimani, M. R. Pournaki and S. Yassemi, Necessary and sufficient conditions for unit graphs to be Hamiltonian, Pacific J. Math. 249 (2) (2011), 419–429. Google Scholar

[12] S. Pirzada, An Introduction to Graph Theory, Universities press, Orient Blakswan, Hyderabad (2012). Google Scholar

[13] S. Pirzada, Bilal A. Rather, Aijaz Ahmad and T. A. Chishti, On distance signless Laplacian spectrum of graphs and spectrum of zero divisor graphs of Zn, Linear Multilinear Algebra (2020) https://doi.org/10.1080/03081087.2020.1838425. Google Scholar

[14] S. Pirzada, Bilal A. Rather and T. A. Chishti, On distance Laplacian spectrum of zero divisor graphs of Zn, Carpathian Math. Publications, 13 (1) (2021), 48–57. Google Scholar

[15] S. Pirzada, Bilal A. Rather, Rezwan Ul Shaban and Merajuddin, On signless Laplacian spectrum of the zero divisor graph of the ring Zn, Korean J. Math. 29 (1) (2021), 13–24. Google Scholar

[16] S. Pirzada, M. Aijaz and Shane Redmond, Upper dimension and bases of zero divisor graphs of commutative rings, AKCE International J. Graphs Comb. 17 (1) (2020), 168–173. Google Scholar

[17] S. Pirzada, Rameez Raja, S. Redmond, Locating sets and numbers of some graphs associated to commutative rings, J. Algebra Appl. 13 (7) (2014), 1450047. Google Scholar

[18] S. Pirzada and Rameez Raja, On the metric dimension of a zero divisor graph, Comm. Algebra 45 (4) (2017), 1399–1408. Google Scholar

[19] Rameez Raja, S. Pirzada and S. Redmond, On locating numbers and codes of zero divisor graphs associated with commutative rings, J. Algebra Appl. 15 (1) (2016), 1650014 (22 pages). Google Scholar

[20] H. Su, K. Noguchi and Y. Zhou, Finite commutative rings with higher genus unit graphs, J. Algebra Appl. 14 (1) (2015), 1550002, 14 pages. Google Scholar

[21] H. Su, G. Tang and Y. Zhou, Rings whose unit graphs are planar, Publ. Math. Debrecen 86 (3-4) (2015), 363–376. Google Scholar

[22] H. Su and Y. Wei, The diameter of unit graphs of rings, Taiwanese J. Math. 23 (1) (2019) 1–10. Google Scholar

[23] H. Su and Y. Zhou, On the girth of the unit graph of a ring, J. Algebra Appl. 13 (2) (2014)1350082, 12 pages. Google Scholar