Some families of graphs whose domination polynomials are unimodal

TL;DR

This paper investigates the unimodality of domination polynomials in graphs, presenting specific graph families where the polynomial exhibits unimodal behavior, thus contributing to the understanding of this conjecture.

Contribution

The paper identifies and proves unimodality of domination polynomials for new families of graphs, advancing the knowledge on the conjecture.

Findings

Certain graph families have unimodal domination polynomials

Supports the conjecture that all domination polynomials are unimodal

Provides new examples and proofs of unimodality

Abstract

Let $G$ be a simple graph of order $n$. The domination polynomial of $G$ is the polynomial $D(G, x)=\sum_{i=\gamma(G)}^{n} d(G,i) x^{i}$, where $d(G,i)$ is the number of dominating sets of $G$ of size $i$ and $\gamma(G)$ is the domination number of $G$. It is conjectured that the domination polynomial of any graph is unimodal. In this paper we present some families of graphs whose domination polynomials are unimodal.