Final title: "More on domination polynomial and domination root" Previous title: "Graphs with domination roots in the right half-plane"

TL;DR

This paper explores the roots of the domination polynomial of graphs, focusing on those with positive real parts, and investigates the polynomial's complexity at specific points for certain graph families.

Contribution

It introduces new families of graphs with domination roots in the right half-plane and analyzes the complexity of the domination polynomial at specific points.

Findings

Identified graph families with domination roots in the right half-plane

Analyzed the complexity of the domination polynomial at specific points

Provided insights into the distribution of domination roots in the complex plane

Abstract

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