The Conjugacy Class and Conjugacy Class Graph of Crystallographic Point Groups of Order 12 and Above
Abstract
In chemistry, crystallographic point group is a set of algebraic groups that maintains at least one point in a fixed position while placing certain restrictions on the rotational symmetries. In this research, the application of group theory and graph theory to the symmetry study of a molecule is presented where the conjugacy classes and conjugacy class graphs of crystallographic point groups of order 12 and above are determined. Conjugacy class is a way of classifying the elements of a group such that two elements a and b are conjugate if there exists an element x and given that xax-1=b. The conjugacy classes obtained are then used to determine the conjugacy class graph in which their vertex set is the set of non-central classes of a group, that is the vertices are connected if the greatest common divisor of the cardinalities of the corresponding vertices is more than one
References
[2] M. Ghorbani, M. Dehmer, S. Rahmani, & M. Rajabi-Parsa, "A Survey on symmetry group of polyhedral graphs". Symmetry, 12(3), 370, 2020.
[3] E. A. Bertram, M. Herzog, & A. Mann, "On a graph related to conjugacy classes of groups", Bulletin of the London Mathematical Society, 22(6), 569-575, 1990.
[4] A. A. Rahman, W. H. Fong, & M. H. A. Hamid, "The conjugacy classes and conjugacy class graphs of point groups of order at most 8". In AIP Conference Proceedings., vol. 1974, no. 1, p. 030007, 2018.
[5] Kong Q. and Wang, S., "A subgraph of conjugacy class graph of finite groups", International Journal of Statistics and Applied Mathematics, 1(3), 34-35, 2016.
[6] Burns G., "Introduction to Group Theory with Applications", Material Science and Technology. Yorktown, New York, Academic Press, 1997.
[7] Ferraro J. R. and Ziomek J. S., "Introductory Group Theory and Its Application to Molecular Structure", Plenum Press, New York, 1995.
[8] Gallian, J., Contemporary abstract algebra. Nelson Education, 2012.
[9] Fraleigh J. B., A First Course in Abstract Algebra. USA: Brooks Cole. 2010.
[10] Bondy J. A. and Murty U. S. R., Graph Theory, London, Springer-Verlag, 2008.
[11] Wilson R. J., and Watkins J. J., Graphs: An Introductory Approach, New York, Springer Science and Business Media, 2003
[12] Bertram, E. A., Herzog, M., & Mann, A., "On a graph related to conjugacy classes of groups", Bulletin of the London Mathematical Society, 22(6), 569-575, 1990.
