Domination polynomial of the commuting and noncommuting graphs of some finite nonabelian groups

Abstract

A dominating set S of a graph is a subset of the vertex set of the graph in which the closed neighborhood of S is the whole vertex set. A domination polynomial of a graph contains coefficients that represent the number of dominating sets in the graph. A domination polynomial is usually being obtained for common types of graphs but not for graphs associated to groups. Two types of graphs associated to groups that are used in this research are the commuting graph and the noncommuting graph. The commuting graph of a group G is a graph whose vertex set contains all noncentral elements of G and any two vertices in the set are adjacent if and only if they commute in G. Meanwhile, the noncommuting graph of a group G is a graph whose vertex set contains all noncentral elements of G and any two vertices in the set are adjacent if and only if they do not commute in G. This paper establishes the domination polynomial of the commuting and noncommuting graphs for the dihedral groups, generalized quaternion groups and quasidihedral groups.