Independence polynomial of the commuting and noncommuting graphs associated to the quasidihedral group

Abstract

An independence polynomial is a type of graph polynomial from graph theory that store combinatorial information such as the graph properties or graph invariants. The independence polynomial of a graph contains coefficients that represent the number of independent sets of certain sizes and the degree of the polynomial denotes the independence number of the graph. A graph of group G is called commuting graph if the vertices are noncentral elements of G and two vertices are adjacent if and only if they commute in G. Meanwhile, a noncommuting graph of a group G has a vertex set that contains all noncentral elements of G and two vertices are adjacent if and only if they do not commute in G. Since the group properties can be presented as graph from graph theory, then the graph polynomial of such graph should also be identified. Therefore, in this research, the independence polynomials are determined for the commuting and noncommuting graphs that are associated to the quasidihedral group.