On the roman domination polynomial of the commuting and non-commuting graphs associated to the dihedral groups

Abstract

A graph associated to a finite group is a way to analyze some properties of a group graphically. Many graphs of groups have been constructed according to the properties of the groups such as the commuting and non-commuting graphs. A Roman dominating function (RDF) of a graph Γ with vertex set V(Γ) and edge set E(Γ) is defined as a function f from the set V(Γ) to the set {0,1,2} such that for any vertex u in V(Γ) with f(u) = 0, there exists at least one vertex v in V(Γ) with f(v) = 2 that is adjacent to u. The sum of the values f(u) for all vertices of Γ is called the weight of the RDF f. The Roman domination number (RDN) of Γ, γ_R(Γ), is the minimum weight among all RDFs defined on Γ. The Roman domination polynomial (RDP) of a graph Γ on p vertices is defined as:

R(Γ, x) = Σ (from j = γ_R(Γ) to 2p) r(Γ,j)x^j

where r(Γ,j) is the number of RDFs of Γ with weight j. In this paper, the RDPs of the commuting and non-commuting graphs associated with dihedral groups of order 2n are computed, and some examples are given to illustrate the results.