A Review on Graphs arising from Finite Groups

Avinash J. Kamble1, Shital Rithe1 and Harshada Pratham1


1Department of Mathematics, Pillai HOC College of Engineering \& Technology, Rasayani, Maharashtra, India.

Abstract: There has been a strong relationship between group (finite) and graph theories for more than a century. Arthur Cayley in 1878 introduced Cayley graphs which geometrically display the action of finite groups. In this paper, we will give a brief description of some specific graphs with its standard results, arising from finite groups.
Keywords: Finite group, Graphs, Vertex-transitive.

Cite this article as: Avinash J. Kamble, Shital Rithe and Harshada Pratham, A Review on Graphs arising from Finite Groups, Int. J. Math. And Appl., vol. 10, no. 1, 2022, pp. 43-49.

References
  1. Arthur Cayley, Desiderata and Suggestions: No.2. The theory of groups: Graphical representation, Amer. J. Math., 1(1878), 174-176.
  2. B. Alspach, Unsolved problem 4.5, Discrete Mathematics, 27(1985).
  3. Alexander Lubotzky, Expander graphs in Pure and Applied Mathematics, Bulletin of the American Mathematical Society, 49(1)(2011), 113–162.
  4. Alexander Lubotzky, Ralph Phillips and Peter Sarnak, Ramanujan graphs, Combinatorica, 8(3)(1988), 261–277.
  5. Avinash J. Kamble and R. Rewaskar, On Structure of Order Graphs Arising from Groups, Annals of Pure and Applied Mathematics, 22(1)(2020), 21-27.
  6. R. Brualdi, Introductory Combinatorics, Fifth Edition, Pearson, (2010).
  7. Béla Bollobás, Random graphs, Cambridge Studies in Advanced Mathematics, Cambridge University Press, (1985).
  8. Biggs Norman, Algebraic Graph Theory, Cambridge University Press, Cambridge, (1993).
  9. Bilal N. Al-Hasanat and Ahmad S. Al-Hasant, Order Graph: A New Representation of finite Groups, International Journal of Mathematics and Computer Science, 14(4)(2019), 809-819.
  10. S. J. Curran and J. A. Gallian, Hamiltonian cycles and paths in Cayley graphs and digraphs - a survey, Discrete Mathematics, 156(1996).
  11. D. B. West, Introduction to Graph Theory, Prentice Hall. Inc., Upper Saddle River, (1996).
  12. Denis Charles, Eyal Goren, and Kristin Lauter, Cryptographic hash functions from expander graphs, Journal of Cryptology, 22(1)(2009), 93–113.
  13. Daniel A. Spielman, Spectral Graph Theory and its Applications, Web, (2011).
  14. D. S. Dummit and R. M. Foote, Abstract Algebra, Third edition, John Wiley and Sons, Inc., Hoboken, (2004).
  15. E. A. Bertram, Some applications of graph theory to finite groups, Discrete Math., 44(1983), 31–43.
  16. E. Konstantinova, Vertex reconstruction in Cayley graphs, Discrete Math., 309(2009), 548–559.
  17. E. Kowalski, Expander Graphs, Web, (2013).
  18. G. R. Pourgholi, H. Yousefi - Azari and A. R. Ashrafi, The Undirected Power Graph of a Finite Group, Bulletin of the Malaysian Mathematical Sciences Society, 38(2015), 1517-1525.
  19. H. E. Rose, A Course on Finite Groups, Cambridge University Press, Cambridge, (1978).
  20. I. M. Isaacs, Finite Group Theory, Graduate Studies in Mathematics, American Mathematical Society, Providence, (2008).
  21. I. Chakrabarty, S. Ghosh and M. K. Sen, Undirected power graphs of semigroups, Semigroup Forum, 78(2009), 410–426.
  22. J. S. Rose, A Course on Group Theory, Dover Publications, Inc., New York, (1994).
  23. M. Mirzargar, A. R. Ashrafi and M. J. Nadjafi-Arani, On the power graph of a finite group, Filomat, 26(6)(2012), 1201–1208.
  24. D. Marusic, Hamiltonian circuits in Cayley graphs, Discrete Mathematics, 46(1983).
  25. Michael A. Nielsen, Introduction to Expander graphs, Web, (2005).
  26. P. J. Cameron and S. Ghosh, The power graph of a finite group, Discrete Mathematics, 311(2011), 1220-1222.
  27. P. J. Cameron, The power graph of a finite group II, Journal of Group Theory, 13(2010), 779–783.
  28. S. H. Payrovi and H. Pasebani, The order Graphs of Groups, Journal of Algebraic Structures and Their Applications, (2014), 1-10.
  29. Shlomo Hoory, Nathan Linial and Avi Wegderson, Expander graphs and their applications, Bulletin of the American Mathematical Society, 43(4)(2006), 439–561.
  30. D. Witte and J. Gallian, A Survey: Hamiltonian cycles in Cayley graphs, Discrete Mathematics, 51(1984).
  31. Yusuf F. Zakariya, Graphs from Finite Groups: An Overview, Proceedings of Annual National Conference, (2016).

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