Characterization of graphs using domination polynomials

Let GG be a simple graph of order nn. The domination polynomial of GG is the polynomial D(G,x)=∑i=1nd(G,i)xi, where d(G,i)d(G,i) is the number of dominating sets of GG of size ii. A root of D(G,x)D(G,x) is called a domination root   of GG. We denote the set of distinct domination roots by Z(D(G,x))Z(D(G,x)). Two graphs GG and HH are said to be DD-equivalent, written as G∼HG∼H, if D(G,x)=D(H,x)D(G,x)=D(H,x). The DD-equivalence class of GG is [G]={H:H∼G}[G]={H:H∼G}. A graph GG is said to be DD-unique if [G]={G}[G]={G}. In this paper, we show that if a graph GG has two distinct domination roots, then Z(D(G,x))={−2,0}Z(D(G,x))={−2,0}. Also, if GG is a graph with no pendant vertex and has three distinct domination roots, then Z(D(G,x))⊆{0,−2±2i,−3±3i2}. Also, we study the DD-equivalence classes of some certain graphs. It is shown that if n≡0,2(mod3), then CnCn is DD-unique, and if n≡0(mod3), then [Pn][Pn] consists of exactly two graphs.