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+%
+% grad.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\begin{frame}[t]
+\frametitle{Grad}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{center}
+\begin{tikzpicture}[>=latex,thick]
+
+\def\r{2.2}
+
+\coordinate (A) at ({\r*cos(0*72)},{\r*sin(0*72)});
+\coordinate (B) at ({\r*cos(1*72)},{\r*sin(1*72)});
+\coordinate (C) at ({\r*cos(2*72)},{\r*sin(2*72)});
+\coordinate (D) at ({\r*cos(3*72)},{\r*sin(3*72)});
+\coordinate (E) at ({\r*cos(4*72)},{\r*sin(4*72)});
+
+\draw[shorten >= 0.2cm,shorten <= 0.2cm] (A) -- (C);
+\draw[color=white,line width=5pt] (B) -- (D);
+\draw[shorten >= 0.2cm,shorten <= 0.2cm] (B) -- (D);
+
+\draw[shorten >= 0.2cm,shorten <= 0.2cm] (A) -- (B);
+\draw[shorten >= 0.2cm,shorten <= 0.2cm] (B) -- (C);
+\draw[shorten >= 0.2cm,shorten <= 0.2cm] (C) -- (D);
+%\draw[shorten >= 0.2cm,shorten <= 0.2cm] (D) -- (E);
+\draw[shorten >= 0.2cm,shorten <= 0.2cm] (E) -- (A);
+
+\draw (A) circle[radius=0.2];
+\draw (B) circle[radius=0.2];
+\draw (C) circle[radius=0.2];
+\draw (D) circle[radius=0.2];
+\draw (E) circle[radius=0.2];
+
+\node at (A) {$1$};
+\node at (B) {$2$};
+\node at (C) {$3$};
+\node at (D) {$4$};
+\node at (E) {$5$};
+\node at (0,0) {$G$};
+
+%\node at ($0.5*(A)+0.5*(B)-(0.1,0.1)$) [above right] {$\scriptstyle 1$};
+%\node at ($0.5*(B)+0.5*(C)+(0.05,-0.07)$) [above left] {$\scriptstyle 2$};
+%\node at ($0.5*(C)+0.5*(D)+(0.05,0)$) [left] {$\scriptstyle 3$};
+%\node at ($0.5*(D)+0.5*(E)$) [below] {$\scriptstyle 4$};
+%\node at ($0.5*(E)+0.5*(A)+(-0.1,0.1)$) [below right] {$\scriptstyle 5$};
+%\node at ($0.6*(A)+0.4*(C)$) [above] {$\scriptstyle 6$};
+%\node at ($0.4*(B)+0.6*(D)$) [left] {$\scriptstyle 7$};
+
+\end{tikzpicture}
+\end{center}
+\begin{block}{Definition}
+Der Grad
+$\deg v$
+eines Knotens $v\in V$ ist die Anzahl der Kanten mit Ende in $v$
+\end{block}
+\end{column}
+\begin{column}{0.48\textwidth}
+\begin{block}{Gradmatrix}
+Diagonalmatrix mit $d_{ii}=\deg v_i$
+\[
+D(G)
+=
+\begin{pmatrix}
+3&0&0&0&0\\
+0&3&0&0&0\\
+0&0&3&0&0\\
+0&0&0&2&0\\
+0&0&0&0&1
+\end{pmatrix}
+\]
+\end{block}
+\begin{block}{Satz}
+Die Summe der Grade ist gerade:
+\[
+\sum_{i=1}^n\deg v_i = \operatorname{Spur} D(G) \equiv 0 \mod 2
+\]
+\end{block}
+\end{column}
+\end{columns}
+\end{frame}