A Survey on Power Graphs of Semigroups

International Journal of Mathematics Trends and Technology (IJMTT)
© 2022 by IJMTT Journal
Volume-68 Issue-8
Year of Publication : 2022
Authors : Miji Philip, P.U. Anusha

How to Cite?

Miji Philip, P.U. Anusha, "A Survey on Power Graphs of Semigroups," International Journal of Mathematics Trends and Technology, vol. 68, no. 8, pp. 105-108, 2022. Crossref, https://doi.org/10.14445/22315373/IJMTT-V68I8P509

The study of the graphical representation of semigroups and groups has become an exciting research area in the past few decades, leading to many captivating results and questions. This article reviews the current state of knowledge on the power graphs of semigroups by presenting all the results recorded in the literature.

Keywords : Semigroup, power graph, Undirected power graph, k- power graph, Enhanced power graph.


[1] J.Abawajy, A. Kelarev, M. Chowdhury, “Power Graphs: A Survey”, Electronic Journal of Graph Theory and Applications, vol. 1, no. 2, pp. 125-147, 2013.
[2] Ajay Kumar, Lavanya Seivaganesh, Peter J. Cameron, T. Tamizh Chelvam, “Recent Developments on the Power Graph of Finite Groups - A Survey,” AKCE International Journal of Graphs and Combinatorics, vol. 18, no. 2, pp. 65-94, 2021.
[3] J. Bosak, “The Graphs of Semigroups,” in Theory of Graphs and Applications, Academic Press, New York, pp. 119-125, 1964.
[4] I. Chakrabarty, S. Ghosh, M. K. Sen, “ Undirected Power Graphs o f Semigroups”, Semigroup Fourm: vol. 78, pp. 410-426, 2009.
[5] S. Chattopadhyay, P. Panigrahi, “Some Structural Properties of Power Graphs and K-Power Graphs of Finite Semigroups”, Journal of Discrete Mathematical Sciences and Cryptography, vol. 20, no. 5, pp. 1101-1119, 2017.
[6] J.M. Howie, “An Introduction to Semigroup Theory”, New York, Academic Press, 1876.
[7] A. V. Kelarev, S. J. Quinn, “Directed Graphs and Combinatorial Properties of Semigroups,” Journal of Algebra, vol. 251, pp. 16-26, 2002.
[8] Sandeep Dalal, Jitender Kumar, Siddharth Singh, “On the Enhanced Power Graph of a Semigroup”, arXiv: 2107.11793v1[math.GR], 2021.
[9] L. John and Padmakumari, “Semigroup Theoretic Study of Cayley graph of Rectangular Bands”, South East Asian Bulletin of Mathematics, vol. 35, pp. 943-950, 2010.
[10] W.B, Vasantha Kandaswamy and Florentin Smarandache, “Semigroups as Graphs,” ZIP Publishing, Ohio, 2012.
[11] A.Riyas and K.Geetha, “A Study on Cayley Graph of Symmetric Inverse Semigroup Relative to Green's Equivalence R-class,” South East Asian Bulletin of Mathematics, vol. 43, no. 1, pp. 133-137, 2019.
[12] A.Riyas, P.U. Anusha and K.Geetha, “A Study of Some Properties of Full Transformation Semigroups,” Springer Proceedings in Mathematics and Statistics, vol. 345.
[13] A.Riyas, P.U.Anusha and K.Geetha, “On Some Properties of Rectangular Band,” Advances in Mathematical Science of Journals, vol. 9, no. 10, pp. 8587-8591, 2020.
[14] A.Riyas, P.U.Anusha and K.Geetha, “On Cayley Graphs of Rees Matrix Semigroup Relative to the Green's Equivalence L-Class,” International Journal of Mathematics and Computer Science, vol. 16, no. 2, pp. 831-835, 2021.
[15] Naveen Palanivel and Chithra A.V, “Some Structural Properties of Unitary Addition Cayley Graphs,” International Journal of Computer and Applications, vol. 121, no. 17, pp. 0975-8887, 2015.
[16] A.V.Kelarev, S.J.Quinn, “A Combinatorial Property of Power Graph of Groups,” Contrib.General Algebra, vol. 12, pp. 229-235, 2000.
[17] A.V.Kelarev , S.J.Quinn, “A Combinatorial Property of Power Graph of Semigroups,” Comment.Math.Univ. Carolin, vol. 45, pp.11-17, 2004.
[18] M.Afkhami, A.Jaferzadeh, K.Khahyarmanesh and S. Mohammadikhah, “On Cyclic Graphs of Finite Semigroups,” J.Algebra Appl., vol. 13, no. 7, pp. 1450035, 2014.
[19] Sandeep Dala, Jitender Kumar and Siddharth Singh, “The Cyclic Graph of a Semigroup,” arXiv:2017.1102v2[math.GR], 2021.
[20] Sandeep Dalal, Jitender Kumar, “Equality of Various Graphs on Finite Semigroups,” arXiv:2007.11376.
[21] Sriparna Chattopadhyay and Prathima Panigrahi, “Some Relations between Power Graphs and Cayley Graphs,” Journal of the Egyptian Mathematical Society, pp. 1110-256X, 2015.
[22] K Ch Das, Night Akgunes and A SinanCevik, “On Monogenic Semigroups,” Journal of Inequalities and Applications, vol. 44, 2013.
[23] N.Agnes, K.C.Das, A.S. Cevik and I.N Hangul, “Some Properties on the Lexicographic Product of Graphs Obtained by Monogenic Semigroups,” Journal of Inequalities and Applications, vol. 238, pp. 1-9, 2013.
[24] Nihat Akgunes, Yasar Nacaroglu and Sedat pak, “Line Graph of Monogenic Semigruop Graphs,” Journal of Mathematics, vol. 2021, pp. 4 , 2021.
[25] Seda Oguz Unal, “An Application of Sombor Index over a special Class of Semigroup,” Journal of Mathematics, vol. 2021.
[26] M. Sattanathan, R.Kala, “An Introduction to Order Prime Graph”, Int. J. Contemp. Math. Science, vol. 4, no. 10, pp. 467-474, 2009.
[27] R. Rajendra, P. Siva Kota Reddy, “On General Order Prime Graph of A Finite Group,” Proceedings of the Jangieon Mathematical Society,vol. 17, no. 4 pp. 641-644, 2014.
[28] X.MA, H.WEI and L.YANG, “The Coprime Graph of a Group”, International Journal of Group Theory, vol. 3, no. 3, pp. 13-23, 2014.