Some new results on domination roots of a graph

TL;DR

This paper investigates the roots of the domination polynomial of graphs, identifying families with no nonzero real roots, and constructs graph families with domination roots dense in the complex plane.

Contribution

It introduces new families of graphs with specific domination root properties and provides a formula for the domination polynomial of the lexicographic product.

Findings

Some graphs have no nonzero real domination roots.

Complex domination roots can have positive real parts.

Domination roots can be dense in the complex plane.

Abstract

Let $G$ be a simple graph of order $n$. The domination polynomial of $G$ is the polynomial $D(G,\lambda)=\sum_{i=0}^{n} d(G,i) \lambda^{i}$, where $d(G,i)$ is the number of dominating sets of $G$ of size $i$. Every root of $D(G,\lambda)$ is called the domination root of $G$. We present families of graphs whose their domination polynomial have no nonzero real roots. We observe that these graphs have complex domination roots with positive real part. Then, we consider the lexicographic product of two graphs and obtain a formula for domination polynomial of this product. Using this product, we construct a family of graphs which their domination roots are dense in all of $\mathbb{C}$.