Domination polynomial of lexicographic product of specific graphs

TL;DR

This paper investigates the domination polynomial of the lexicographic product of specific graphs, providing formulas and properties that enhance understanding of domination in complex graph structures.

Contribution

It introduces new formulas for the domination polynomial of the lexicographic product of certain graphs, expanding theoretical knowledge in graph domination polynomials.

Findings

Derived explicit formulas for domination polynomials of specific graph products

Identified properties and behaviors of domination polynomials in these products

Enhanced understanding of domination in complex graph structures

Abstract

Let $G$ be a simple graph of order $n$. The domination polynomial of $G$ is the polynomial $D(G,\lambda)=\sum_{i=0}^{n} d(G,i) \lambda^{i}$, where $d(G,i)$ is the number of dominating sets of $G$ of size $i$. We consider the lexicographic product of two specific graphs and study their domination polynomials.