03b9aa91-65ca-4ad1-9769-132c4178692720210830085046487wseas:wseasmdt@crossref.orgMDT DepositWSEAS TRANSACTIONS ON MATHEMATICS2224-28801109-276910.37394/23206http://wseas.org/wseas/cms.action?id=40513220213220212010.37394/23206.2021.20https://wseas.org/wseas/cms.action?id=23278Mohammad HassanMudaberDepartment of Mathematical Sciences, Faculty of Science Universiti Teknologi Malaysia 81310 UTM Johor Bahru, MALAYSIANor HanizaSarminDepartment of Mathematical Sciences, Faculty of Science Universiti Teknologi Malaysia 81310 UTM Johor Bahru, MALAYSIAIbrahimGamboDepartment of Mathematical Sciences, Faculty of Science Universiti Teknologi Malaysia 81310 UTM Johor Bahru, MALAYSIAThe induced subgraph of a unit graph with vertex set as the idempotent elements of a ring R is a graph which is obtained by deleting all non idempotent elements of R. Let C be a subset of the vertex set in a graph Γ. Then C is called a perfect code if for any x, y ∈ C the union of the closed neighbourhoods of x and y gives the the vertex set and the intersection of the closed neighbourhoods of x and y gives the empty set. In this paper, the perfect codes in induced subgraphs of the unit graphs associated with the ring of integer modulo n, Zn that has the vertex set as idempotent elements of Zn are determined. The rings of integer modulo n are classified according to their induced subgraphs of the unit graphs that accept a subset of a ring Zn of different sizes as the perfect codes8302021830202139940341https://wseas.com/journals/mathematics/2021/a825106-016(2021).pdf10.37394/23206.2021.20.41https://wseas.com/journals/mathematics/2021/a825106-016(2021).pdfC. E. Shannon, A mathematical theory of communication, The Bell system technical journal, Vol. 27, No. 3, 1948, pp. 379­423. 10.1002/j.1538-7305.1950.tb00463.xR. W. Hamming, Error detecting and error correcting codes, The Bell system technical journal, Vol. 29, No. 2, 1950, pp. 147­160. W. K. Nicholson, Introduction to abstract algebra, John Wiley Sons, 2012. 10.1216/rmj-1975-5-2-199J. H. Van Lint, A survey of perfect codes, The Rocky Mountain Journal of Mathematics, Vol. 5, No. 2, 1975, pp. 199­224. N. Biggs, Perfect codes in graphs, Journal of Combinatorial Theory, Series B, Vol. 15, No. 3, 1973, pp. 289­296. J. Kratochvil, Perfect codes over graphs, Journal of Combinatorial Theory, Series B, Vol. 40, No. 2, 1986, pp. 224­228. 10.1137/17m1129532H. Huang, B. Xia, and S. Zhou, Perfect codes in Cayley graphs, SIAM Journal on Discrete Mathematics, Vol. 32, No. 1, 2018, pp. 548­559. 10.1137/19m1258013X. Ma, G. L. Walls, K. Wang, and S. Zhou, Subgroup perfect codes in Cayley graphs, SIAM Journal on Discrete Mathematics, Vol. 34, No. 3, 2020, pp. 1909­1921. J. Zhang and S. Zhou, On subgroup perfect codes in Cayley graphs, European Journal of Combinatorics, Vol. 91, 2021, p. 103228. 10.1016/j.disc.2020.111813J. Chen, Y. Wang, and B. Xia, Characterization of subgroup perfect codes in Cayley graphs, Discrete Mathematics, Vol. 343, No. 5, 2021, p. 111813. 10.1080/00927870903095574N. Ashrafi, H. Maimani, M. Pournaki, and S. Yassemi, Unit graphs associated with rings, Communications in Algebra, Vol. 38, No. 8, 2010, pp. 2851­2871. H. Su and Y. Zhou, On the girth of the unit graph of a ring, Journal of Algebra and Its Applications, Vol. 13, No. 2, 2014, pp. 1­12. 10.5486/pmd.2015.6096H. Su, G. Tang, and Y. Zhou, Rings whose unit graphs are planar, Publ. Math. Debrecen, Vol. 86, No. 3­4, 2015, pp. 363­376. 10.4171/rsmup/133-9S. Kiani, H. Maimani, M. Pournaki, and S. Yassemi, Classification of rings with unit graphs having domination number less than four, Rend. Sem. Mat. Univ. Padova, Vol. 133, 2015, pp. 173­195. H. Su and Y. Wei, The diameter of unit graphs of rings, Taiwanese Journal of Mathematics, Vol. 23, No. 1, 2019, pp. 1­10. 10.2140/pjm.2011.249.419H. R. Maimani, M. Pournaki, and S. Yassemi, Necessary and sufficient conditions for unit graphs to be Hamiltonian, Pacific journal of mathematics, Vol. 249, No. 2, 2011, pp. 419­ 429.