Characterization of graphs using domination polynomials

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