Graph polynomial associated to some graphs of certain finite nonabelian groups

Abstract

The study of associating the groups in group theory with the graphs in graph theory are widely done by many researchers. Since the algebraic properties of groups can be studied through the structures of graphs, then it is common to find certain graph invariants and graph properties. Nevertheless, the graph polynomials are also significant in the study of the graphs but have not yet determined for the graphs associated to groups. Graph polynomials, such as the independence polynomial, the clique polynomial, and the domination polynomial are used to store the combinatorial information of a graph. An independence polynomial of a graph is the polynomial in which its coefficients are the number of independent sets in the graph. A clique polynomial of a graph is the polynomial containing coefficients that represent the number of cliques in the graph. Meanwhile, a domination polynomial of a graph is the polynomial that contains coefficients representing the number of dominating sets in the graph. In the first part of this research, these three polynomials are determined for five types of graphs for three types of groups. The graphs considered are the conjugacy class graphs, the conjugate graphs, the commuting graphs, the noncommuting graphs, and the center graphs associated to the dihedral group, the generalized quarternion group, and the quasidihedral group. All these graphs are found and expressed in general in the form of the union and join of some complete graphs, complete bipartite graphs, and also complete multipartite graphs. Then, the graph polynomials associated to groups are obtained from these common types of graphs by using the properties of the graph polynomials. The results obtained are some polynomials of certain degrees. In the second part of this research, the roots of all the graph polynomials associated to the finite groups that have been computed are determined. The independence polynomial of the graphs associated to groups have real roots that are always negative. The clique polynomials have roots that are always real but may not be integers. Meanwhile, the domination polynomials always have a zero root and the other roots may be complex numbers. In the last part of this research, two types of new graph polynomials are defined and determined for the graphs mentioned earlier. The new graph polynomials are called the clique-independence polynomial and the clique-domination polynomial. The clique-independence polynomial of a graph is the polynomial containing coefficients that represent the number of clique- independent sets in the graph. The clique-domination polynomial of a graph is the polynomial in which its coefficients are the number of clique-dominating sets in the graph. The clique-independence polynomials are obtained for the conjugacy class graph, the conjugate graph, the commuting graph, the noncommuting graph, and the center graph associated to the dihedral group because these graphs contain clique- independent sets and are suitable to be expressed in the form of clique-independence polynomials. Meanwhile, the clique-domination polynomials are determined only for the noncommuting graph associated to the dihedral group since clique-dominating sets exist only for connected graphs.