Some families of graphs with no nonzero real domination roots

TL;DR

This paper investigates specific families of graphs that lack nonzero real roots in their domination polynomials, contributing to understanding the algebraic properties of graph domination.

Contribution

It identifies and presents new families of graphs with the property that their domination polynomials have no nonzero real roots.

Findings

Certain graph families have domination polynomials with only zero as a real root.

The study expands knowledge on the algebraic structure of domination polynomials.

Results may influence future research on graph polynomial roots.

Abstract

Let G be a simple graph of order n. The domination polynomial is the generating polynomial for the number of dominating sets of G of each cardinality. A root of this polynomial is called a domination root of G. Obviously 0 is a domination root of every graph G. In the study of the domination roots of graphs, this naturally raises the question: which graphs have no nonzero real domination roots? In this paper we present some families of graphs whose have this property.