The independence and clique polynomials of the center graphs of some finite groups

Abstract

The independence polynomial and the clique polynomial are the graph poly- nomials that are used to describe the combinatorial information of graphs, including the graphs related to group theory. An 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. A clique poly- nomial of a graph is the polynomial containing coefficients that represent the number of cliques in the graph. The clique of a graph is a set of vertices that are adjacent to each other in the graph. Meanwhile, the center graph of a group G is a graph in which the vertices are all the elements of G and two distinct vertices a, b are adjacent if an only if ab is in the center of G. In this research, the independence polynomial and the clique polynomial are established for the center graphs of three finite non- abelian groups, namely the dihedral group, the generalized quarternion group and the quasidihedral group.