On the Unimodality of Domination Polynomials

TL;DR

This paper proves that domination polynomials of paths, cycles, and complete multipartite graphs are unimodal, and that most graphs have unimodal domination polynomials with a mode near half their size.

Contribution

It establishes the unimodality of domination polynomials for several classes of graphs and shows that almost all graphs have unimodal domination polynomials with a specific mode.

Findings

Domination polynomials of paths, cycles, and complete multipartite graphs are unimodal.

Almost every graph has a unimodal domination polynomial.

The mode of the domination polynomial for most graphs is near half the number of vertices.

Abstract

A polynomial is said to be unimodal if its coefficients are non-decreasing and then non-increasing. The domination polynomial of a graph $G$ is the generating function of the number of domination sets of each cardinality in $G$, and its coefficients have been conjectured to be unimodal. In this paper we will show the domination polynomial of paths, cycles and complete multipartite graphs are unimodal, and that the domination polynomial of almost every graph is unimodal with mode $ \lceil \frac{n}{2}\rceil $.