Some families of graphs with no nonzero real domination roots

Abstract

Let G be a simple graph of order n. The domination polynomial of G is the polynomial D(G, x) = ∑ⁿ_{i = γ(G)}d(G, i)xⁱ, where d(G, i) is the number of dominating sets of G of size i and γ(G) is the domination number of G. A root of D(G, x) is called a domination root of G. Obviously, 0 is a domination root of every graph G with multiplicity γ(G). In the study of the domination roots of graphs, this naturally raises the question: Which graphs have no nonzero real domination roots? In this paper we present some families of graphs whose have this property.