# The independence and clique polynomial of the conjugacy class graph of dihedral group

## DOI:

https://doi.org/10.11113/mjfas.v14n0.1268## Keywords:

Independence polynomial, clique polynomial, conjugacy class graph, dihedral group## Abstract

The independence and clique polynomial are two types of graph polynomial that store combinatorial information of a graph. The independence polynomial of a graph is the polynomial in which its coefficients are the number of independent sets in the graph. The independent set of a graph is a set of vertices that are not adjacent. The clique polynomial of a graph is the polynomial in which its coefficients are the number of cliques in the graph. The clique of a graph is a set of vertices that are adjacent. Meanwhile, a graph of group *G* is called conjugacy class graph if the vertices are non-central conjugacy classes of *G* and two distinct vertices are connected if and only if their class cardinalities are not coprime. The independence and clique polynomial of the conjugacy class graph of a group *G* can be obtained by considering the polynomials of complete graph or polynomials of union of some graphs. In this research, the independence and clique polynomials of the conjugacy class graph of dihedral groups of order *2n* are determined based on three cases namely when *n* is odd, when *n* and *n/2* are even, and when *n* is even and *n/2* is odd. For each case, the results of the independence polynomials are of degree two, one and two, and the results of the clique polynomials are of degree *(n-1)/2*, *(n+2)/2 *and *(n-2)/2*, respectively.

## References

Balakrishnan, R. and Ranganathan, K. 2012. A Textbook of Graph Theory. 2nd ed. New York: Springer.

Rosen, K. H. 2013. Discrete Mathematics and Its Applications. 7th ed. New York: McGraw-Hill

Hoede, C. and Li, X. 1994. Clique Polynomials and Independent Set Polynomials of Graphs. Discrete Mathematics. 125: 219-228.

Ferrin, G. 2014. Independence Polynomials. Master Dissertation. University of California.

Harvey, E. R. 2009. A Course in Finite Groups. London: Springer-Verlag.

Samaila, D., Abba, B. I. and Pur, M. P.. 2013. On the conjugacy classes, centers and representation of the groups and . International Journal of Pure and Applied Science and Technology. 15(1): 87-95.

Bertram, E. A., Herzog, M. and Mann, A. 1990. On a graph related to conjugacy classes of groups. Bulletin of the London Mathematical Society. 22(6): 569-575.

Mahmoud, R., Sarmin, N. H. and Erfanian, A. 2017. The conjugacy class graph of some finite groups and its energy. Malaysian Journal of Fundamental and Applied Sciences. 13(4): 659-665.

Omer, S. M. S., Sarmin, N. H. and Erfanian, A.. 2013. The probability that an element of a group fixes a set and its graph related to conjugacy classes. Journal of Basic Applied Scientific Research. 3(10): 369-380.

Sarmin, N. H., Gambo, I. and Omer, S. M. S. 2015. The conjugacy classes of metabelian groups of order at most 24. Jurnal Teknologi 77(1) : 139-143.

Sarmin, N. H., Bilhikmah, N. H., Omer, S. M. S. and Noor, A. H. M. 2016. The conjugacy classes, conjugate graph and conjugacy class graph of some finite metacyclic 2-groups. Menemui Matematik. 38(1): 1-12.