On the roots of domination polynomial of graphs

Abstract

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)=∑i=1nd(G,i)xi, 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 δ=δ(G) be the minimum degree of vertices of G. We prove that all roots of D(G,x) lies in the set {z:|z+1|≤2n−1δ+1}. We show that D(G,x) has at least δ−1 non-real roots. In particular, we prove that if all roots of D(G,x) are real, then δ=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 δ≥2n3−1, 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.