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