On the Real Roots of Domination Polynomials

TL;DR

This paper investigates the real roots of domination polynomials in graphs, proving that their closure is the interval from negative infinity to zero, thus characterizing their root distribution.

Contribution

It establishes that the closure of the real roots of domination polynomials is exactly the interval (-∞, 0], providing a complete characterization.

Findings

Real roots of domination polynomials are contained in (-∞, 0]

Closure of these roots is exactly (-∞, 0]

Provides insight into the root distribution of domination polynomials

Abstract

A dominating set $S$ of a graph $G$ of order $n$ is a subset of the vertices of $G$ such that every vertex is either in $S$ or adjacent to a vertex of $S$. The domination polynomial is defined by $D(G,x) = \sum d_k x^k$ where $d_k$ is the number of dominating sets in $G$ with cardinality $k$. In this paper we show that the closure of the real roots of domination polynomials is $(-\infty,0]$.