Graph polynomials of the conjugate graph of dihedral groups

Abstract

There are various aspects of combinatorial information that are stored in the coefficients of graph polynomials such as the independence polynomial, the matching polynomial and the clique polynomial. The independence polynomial of a graph is defined as a polynomial in which its coefficients are the number of independent sets in the graph. The independent set of a graph is a set of pairwise non-adjacent vertices. The matching polynomial of a graph is the polynomial in which the coefficients are the number of matching sets in the graph. The matching set of a graph is a set of pairwise edges which do not have common vertices. The clique polynomial of a graph is the polynomial in which the coefficients are the number of cliques in the graph. The clique of a graph is the set of pairwise adjacent vertices. Meanwhile, a graph of group G is called conjugate graph if the vertices are non-central elements of G and two distinct vertices are connected if they are conjugate. In this research, the independence polynomial, the matching polynomial and the clique polynomial of the conjugate graph of dihedral groups of order at most twelve are computed.