On the roots of domination polynomial of graphs

Let G be a graph of order n. A dominating set of G is a subset of vertices of G, say S, such that every vertex in V(G) \ S is adjacent to at least one vertex of S. The domination polynomial of G is the polynomial D(G, x) = Sigma(n)(i=1) 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. Let delta = delta(G) be the minimum degree of vertices of G. We prove that all roots of D(G, x) lies in the set {z : vertical bar z + 1 vertical bar <= (delta+1)root 2(n) - 1 . We show that D(G, x) has at least delta - 1 non-real roots. In particular, we prove that if all roots of D(G, x) are real, then delta = 1. We construct an infinite family of graphs such that all roots of their polynomials are real. Motivated by a conjecture (Akbari, et al. 2010) which states that every integer root of D(G, x) is -2 or 0, we prove that if 8 >= 2n/3 - 1, then every integer root of D(G, x) is -2 or 0. Also we prove that the conjecture is valid for trees and unicyclic graphs. Finally we characterize all graphs that their domination roots are integer. (C) 2015 Elsevier B.V. All rights reserved.