## DOI

https://doi.org/10.25772/0J9C-G563

## Defense Date

2010

## Document Type

Thesis

## Degree Name

Master of Science

## Department

Mathematical Sciences

## First Advisor

Craig Larson

## Abstract

A benzenoid is a molecule that can be represented as a graph. This graph is a fragment of the hexagon lattice. A dominating set $D$ in a graph $G$ is a set of vertices such that each vertex of the graph is either in $D$ or adjacent to a vertex in $D$. The domination number $\gamma=\gamma(G)$ of a graph $G$ is the size of a minimum dominating set. We will find formulas and bounds for the domination number of various special benzenoids, namely, linear chains $L(h)$, triangulenes $T_k$, and parallelogram benzenoids $B_{p,q}$. The domination ratio of a graph $G$ is $\frac{\gamma(G)}{n(G)}$, where $n(G)$ is the number of vertices of $G$. We will use the preceding results to prove that the domination ratio is no more than $\frac{1}{3}$ for the considered benzenoids. We conjecture that is true for all benzenoids.

## Rights

© The Author

## Is Part Of

VCU University Archives

## Is Part Of

VCU Theses and Dissertations

## Date of Submission

May 2010