# The independence polynomial of n-th central graph of dihedral groups

## DOI:

https://doi.org/10.11113/mjfas.v13n3.550## Keywords:

Independence polynomial, n-th central graph, dihedral group## Abstract

An independent set of a graph is a set of pairwise non-adjacent vertices while the independence number is the maximum cardinality of an independent set in the graph. The independence polynomial of a graph is defined as a polynomial in which the coefficient is the number of the independent set in the graph. Meanwhile, a graph of a group *G* is called *n*-th central if the vertices are elements of *G* and two distinct vertices are adjacent if they are elements in the *n*-th term of the upper central series of *G*. In this research, the independence polynomial of the *n*-th central graph is found for some dihedral groups.

## References

Balakrishnan, P., Sattanathan, M., Kala, R. 2011. The center graph of a group. South Asian Journal of Mathematics. 1(1), 21-28.

Balakrishnan, R. and Ranganathan, K. 2012. A textbook of graph theory, 2nd ed. New York: Springer.

Bertram, E. A. 1983. Some applications of graph theory to finite groups. Discrete Mathematics. 44, 31-43.

Ferrin, G. 2014. Independence Polynomials, Master Dissertation.

Fraleigh, J. B. 2003. A first course in abstract algebra, 7th ed. U.S.A.: Pearson Education, Inc.

Hoede, C. and Li, X. 1994. Clique polynomials and independent set polynomials of graphs. Discrete Mathematics. 125, 219-228.

Karimi, Z., Erfanian, A., Tolue, B. 2016. n-th central graph of a group. Comptes rendus de l’Acade'mie bulgare des Sciences. 69, 135-144.

Levit, V. E. and E. Mandrescu. 2005. The Independence Polynomial of a Graph – A Survey. Proceedings of the 1st International Conference on Algebraic Informatics. Aristotle Univ.

Thessaloniki. 3 October 2005. Thessaloniki. 233254.

Rose, H. E. 2009. A course on finite groups. London: Springer-Verlag.

Rosen, K. H. 2013. Discrete mathematics and its applications, 7th ed. New York: McGraw-Hill.

Rotman, J. J. 2003. Advanced modern algebra. USA: Prentice Hall.