The Perfect Codes of Non-coprime and Coprime Graphs
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Authors: B. TOLUE AND A. ERFANIAN
DOI: 10.46793/KgJMat2506.913T
Abstract:
In this paper, we focus on the perfect and total perfect codes of the non-coprime and coprime graphs associated to the dihedral groups and finite Abelian groups. We used the advantage of independent sets and tried to present the independent polynomial for them.
Keywords:
Coprime graph, non-coprime graph, perfect code, total perfect code.
References:
[1] N. Biggs, Perfect codes in graphs, J. Combin. Theory Ser. B 15 (1973), 289–296. http://dx.doi.org/10.1016/0095-8956(73)90042-7
[2] H. Dorbidi, A note on the coprime graph of a group, Int. J. Group Theory 5(4) (2016), 17–22.
[3] A-A. Ghidewon, R. H. Hammack and D. T. Taylor, Total perfect codes in tensor products of graphs, Ars Combin. 88 (2008), 129–134.
[4] J. Kratochví, Perfect codes over graphs, J. Combin. Theory Ser. B 40(2) (1986), 224–228. http://dx.doi.org/10.1016/0095-8956(86)90079-1
[5] S. Ling and C. Xing, Coding Theory a First Course, Cambridge University Press, 2004.
[6] X. Ma, Perfect codes in proper reduced power graphs of finite groups, Commun. Algebra 48(9) (2020), 3881–3890. http://dx.doi.org/10.1080/00927872.2020.1749845
[7] X. Ma, G. L. Walls, K. Wang and S. Zhou, Subgroup perfect codes in Cayley graphs, SIAM J. Discrete Math. 34(3) (2020), 1909–1921. http://dx.doi.org/10.1137/19M1258013
[8] X. Ma, H. Wei and L. Yang, The coprime graph of a group, Int. J. Group Theory 3(3) (2014), 13–23.
[9] F. Mansoori, A. Erfanian and B. Tolue, Non-coprime graph of a finite group, AIP Conference Proceedings 1750(1) (2016), Article ID 050017. https://doi.org/10.1063/1.4954605
[10] C. E. Shannon, A mathematical theory of communication, The Bell System Technical Journal 27 (1948), 379–423, 623–656. http://dx.doi.org/10.1002/j.1538-7305.1948.tb01338.x, http://dx.doi.org/10.1002/j.1538-7305.1948.tb00917.x