diff --git a/presentation/Makefile b/presentation/Makefile
index 6edcd91..5942e41 100644
--- a/presentation/Makefile
+++ b/presentation/Makefile
@@ -2,16 +2,21 @@ latex=xelatex
 pdflatex=xelatex
 bibtex=bibtex 
 
+graph_pngs= graphs/valid_graph.png
+
+
 
 all: main.pdf
 
-main.pdf: main.tex #main.bib $(cover) $(chapters)
+main.pdf: main.tex $(graph_pngs)
 	$(latex) main
 	#$(bibtex) main
 	$(latex) main
 	$(pdflatex) main
 
 
+graphs/%.png: graphs/%.dot
+	dot $< -Tpng -o $@
 	
 clean:
 	-rm main.aux
@@ -22,3 +27,4 @@ clean:
 	-rm main.pdf
 	-rm main.toc
 	-rm main.bbl
+	-rm $(graph_pngs)
diff --git a/presentation/graphs/valid_graph.dot b/presentation/graphs/valid_graph.dot
new file mode 100644
index 0000000..004df18
--- /dev/null
+++ b/presentation/graphs/valid_graph.dot
@@ -0,0 +1,8 @@
+graph default
+{
+    0 -- 1 -- 2
+    1 -- 4 -- 2
+    0 -- 2 -- 5
+    6 -- 7 -- 8
+    3
+}
diff --git a/presentation/main.tex b/presentation/main.tex
index 7fab365..2bd91e5 100644
--- a/presentation/main.tex
+++ b/presentation/main.tex
@@ -13,6 +13,7 @@
 %\usepackage{struktex}
 \usepackage{qcircuit}
 \usepackage{adjustbox}
+\usepackage{tikz}
 
 
 \usetheme{metropolis}
@@ -500,4 +501,75 @@
 \end{frame}
 }
 
+\section{Graphical Description of Stabilizer States}
+
+{
+\begin{frame}{Graphs}
+    \begin{itemize}
+        \item{\textbf{Definition}
+            {\itshape
+            The tuple $(V, E)$ is called a graph iff $V$ is a set of vertices with $|V| = n \in \mathbb{N}$ elements.
+            In the following $V = \{0, ..., n-1\}$ will be used.
+            $E$ is the set of edges $E \subset \left\{\{i, j\} \middle| i,j \in V, i \neq j\right\}$.
+            }}
+        \item{
+            Example for a valid graph:\\
+            \includegraphics[width=\linewidth,height=0.5\textheight,keepaspectratio]{graphs/valid_graph.png}
+            }
+    \end{itemize}
+
+\end{frame}
+}
+
+{
+\begin{frame}{ VOP-free Graph States}
+    \begin{itemize}
+        \item{\textbf{Definition}
+            {\itshape
+            For $G = (V,E)$, $i \in V$ define 
+            \begin{equation}
+            K_G^{(i)} := X_i \prod\limits_{\{i,j\} \in E} Z_j
+            \end{equation}
+            the stabilizers associated with the graph $G$.
+            }}
+        \item{
+            The state stabilized by all $K_G^{(i)}$ is 
+            \begin{equation}
+                \ket{\bar{G}} = \prod\limits_{\{i,j\} \in E} CZ_{i,j} \ket{+}.
+            \end{equation}
+            This state is called vertex operator-free (VOP-free) graph state.
+            }
+        \item{
+                Applying a $CZ_{i,j}$ gate toggles the edge $\{i,j\}$ in $E$.
+            }
+    \end{itemize}
+\end{frame}
+}
+
+{
+\begin{frame}{Dynamics of VOP-free Graph States}
+    \begin{itemize}
+        \item{
+            For $a \in V$ the transformation 
+            \begin{equation}
+                M_a := \sqrt{-iX_i} \prod\limits_{\{i,j\} \in E} \sqrt{iZ_j}
+            \end{equation}
+            toggles the neighbourhood $n_a := \left\{ j \middle| \{a,j\} \in E\right\}$
+            of a.
+            }
+        \item{
+            Many Clifford operations cannot be described by the VOP-free graph states.\\
+            Example:
+            \begin{equation}
+                G = \left(\{0, 1\}, \{\}\right)
+                %\ket{\bar{G}} &= \ket{+}\\
+                %U &= H_0H_1 \\
+                %U \ket{\bar{G}} &= \ket{\mbox{0b}00}\\
+            \end{equation}
+
+
+            }
+    \end{itemize}
+\end{frame}
+}
 \end{document}