On the conjugate graphs of finite p-groups
DOI:
https://doi.org/10.11113/mjfas.v13n2.557Keywords:
Group TheoryAbstract
Graphs can be related to groups by looking at its vertices and edges. The vertices are comprised of the elements or sets from the groups and the edges are the properties and conditions for the graph. Recently, research on graphs of groups have attracted many authors. A conjugate graph of a group is defined as; its vertex set is the set of non-central classes of G, and two distinct vertices A and B are connected by an edge if and only if they are conjugate. In this research, the conjugacy class of some finite p-groups are first found. Then, the conjugate graph are determined.
References
Laffey, T. J. 1978. Centralizers of elementary abelian subgroups in finite p-groups. Journal of Algebra. 51(1), 88-96.
Zainal, R., Mohd Ali, N. M., Sarmin, N. H., & Rashid, S. 2013. On the non-abelian tensor square of groups of order p4 where p is an odd prime. Science Asia 39S, 16-8.
Rashid, S., Nawi, A. A., Zainal, R., Mohd Ali, N. M., & Sarmin, N. H. 2014. The use of Groups, Algorithms and Programming (GAP) software in calculating the commutator subgroup and centre of groups of order p3q. Science Asia. 40S, 73-77.
Erfanian, A., Tolue, B. 2012. Conjugate graphs of finite groups. Discrete Mathematics, Algorithms and Applications. 4(2), 1250035.
Bertram, E. A., Herzog, M., & Mann, A. 1990. On a graph related to conjugacy classes of groups. Bulletin of the London Mathematical Society. 22(6), 569-575.
Segev, Y. 2001. The commuting graph of minimal nonsolvable groups. Geometriae Dedicata. 88(1), 55-66.
Akbari, S., Mohammadian, A., Radjavi, H., Raja, P. 2006. On the diameters of commuting graphs. Linear Algebra and its Applications. 418(1), 161-176.
Bundy, D. 2006. The connectivity of commuting graphs. Journal of Combinatorial Theory, Series A. 113(6), 995-1007.
Darafsheh, M. R. 2009. Groups with the same non-commuting graph. Discrete applied mathematics. 157(4), 833-837.
Omer, S. M. S., Moradipour, K., Erfanian, A. 2013. The Probability That an Element of a Group Fixes a Set and Its Graph Related to Conjugacy Classes. Journal of Basic and Applied Scientific Research. 3(10), 369-380.
Moradipour, K., Sarmin, N. H., Erfanian, A. 2013. On Non-commuting Graphs of Some Finite Groups. International Journal of Applied Mathematics and Statistics. 45(15), 473-476.
Sarmin, N. H., El-sanfaz, M. A., Omer, S. M. S. 2016. Groups and graphs in probability theory. In AIP Conference Proceedings. 1750: 50012.
Darafsheh, M. R., Bigdely, H., Bahrami, A., Monfared, M. D. 2010. Some results on non-commuting graph of a finite group. Italian Journal of Pure and Applied Mathematics. 27, 107-118.
Rotman, J. J. 2002. Advanced Modern Algebra. USA: Pearson Education.
Xiong, B. and Zheng, Z. 2010. Graph Theory. Shanghai, China: East China Normal University Press.
Diestel, R. 2005. Graph Theory. (3rd Edition). Germany: Springer.
Wilson, R. J. 1972. Introduction to Graph Theory. USA: Academic Press.
Burnside,W. 1911. Theory of groups of finite order. (2nd Edition). Cambridge: Cambridge University Press.
