Domination polynomials of k-tree related graphs

TL;DR

This paper investigates the domination polynomials of various k-tree related graphs, analyzing their properties and identifying families with domination polynomials that lack nonzero real roots.

Contribution

It introduces new results on the domination polynomials of k-tree related graphs and identifies specific families with unique root properties.

Findings

Domination polynomials of certain k-tree related graphs are characterized.

Some graph families have domination polynomials with no nonzero real roots.

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 $G$. In this paper we study the domination polynomials of several classes of $k$-tree related graphs. Also, we present families of these kind of graphs, whose domination polynomial have no nonzero real roots.