Summary: Let be a simple graph of order . The domination polynomial of is the polynomial
where is the number of dominating sets of of size . A root of is called a domination root of . We denote the set of distinct domination roots by . Two graphs and are said to be -equivalent, written as , if . The -equivalence class of is . A graph is said to be -unique if .
In this paper, we show that if a graph has two distinct domination roots, then . Also, if is a graph with no pendant vertex and has three distinct domination roots, then
Also, we study the -equivalence classes of some certain graphs. It is shown that if , then is -unique, and if , then consists of exactly two graphs.