More on Domination Polynomial and Domination Root

Abstract

Let $G$ be a simple graph of order $n$. The domination polynomial of $G$ is the polynomial $D(G,x)=\sum _{i=0}^{n}d(G,i){\lambda }^i$, where $d(G,i)$ is the number of dominating sets of $G$ of size $i$. Every root of $D(G,\lambda )$ is called a domination root of $G$. It is clear that $(0,\mathrm{\infty })$ is a zero-free interval for the domination polynomial of a graph. It is interesting to investigate graphs that have complex domination roots with positive real parts. In this paper, we first investigate the complexity of the domination polynomial at specific points. Then, we present and investigate some families of graphs whose complex domination roots have positive real parts.