An atlas of domination polynomials of graphs of order at most six

TL;DR

This paper systematically computes and tabulates the domination polynomials and roots for all graphs with up to six vertices, providing a comprehensive reference for graph domination properties.

Contribution

It presents the first complete enumeration of domination polynomials and roots for all graphs of order at most six, filling a gap in graph theory literature.

Findings

Complete tables of domination polynomials for all graphs up to six vertices.

Identification of domination roots for small graphs.

Foundation for further research on domination polynomial properties.

Abstract

The domination polynomial of a graph $G$ of order $n$ 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 $G$. The roots of domination polynomial is called domination roots. In this article, we compute the domination polynomial and domination roots of all graphs of order less than or equal to 6, and show them in the tables.