add new slides

4 files changed, 219 insertions, 0 deletions

diff --git a/vorlesungen/slides/8/Makefile.inc b/vorlesungen/slides/8/Makefile.inc
index e8c7502..903fce2 100644
--- a/vorlesungen/slides/8/Makefile.inc
+++ b/vorlesungen/slides/8/Makefile.inc
@@ -7,5 +7,7 @@
 chapter8 =								\
 	../slides/8/dgraph.tex						\
 	../slides/8/graph.tex						\
+	../slides/8/grad.tex						\
+	../slides/8/inzidenz.tex					\
 	../slides/8/chapter.tex
 
diff --git a/vorlesungen/slides/8/chapter.tex b/vorlesungen/slides/8/chapter.tex
index 761ea63..23d9aac 100644
--- a/vorlesungen/slides/8/chapter.tex
+++ b/vorlesungen/slides/8/chapter.tex
@@ -5,3 +5,5 @@
 %
 \folie{8/graph.tex}
 \folie{8/dgraph.tex}
+\folie{8/grad.tex}
+\folie{8/inzidenz.tex}
diff --git a/vorlesungen/slides/8/grad.tex b/vorlesungen/slides/8/grad.tex
new file mode 100644
index 0000000..a232828
--- /dev/null
+++ b/vorlesungen/slides/8/grad.tex
@@ -0,0 +1,84 @@
+%
+% 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}
diff --git a/vorlesungen/slides/8/inzidenz.tex b/vorlesungen/slides/8/inzidenz.tex
new file mode 100644
index 0000000..5911593
--- /dev/null
+++ b/vorlesungen/slides/8/inzidenz.tex
@@ -0,0 +1,131 @@
+%
+% inzidenz.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\definecolor{darkgreen}{rgb}{0,0.6,0}
+\begin{frame}[t]
+\frametitle{Inzidenz- und Adjazenzmatrix}
+\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[color=darkgreen,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);
+
+\fill[color=red!20] (B) circle[radius=0.2];
+
+\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 ($0.5*(E)+0.5*(A)+(-0.1,0.1)$) [below right] {$\scriptstyle 4$};
+\node at ($0.6*(A)+0.4*(C)$) [above] {$\scriptstyle 5$};
+\node[color=darkgreen] at ($0.4*(B)+0.6*(D)$) [left] {$\scriptstyle 6$};
+
+\end{tikzpicture}
+\end{center}
+\vspace{-10pt}
+\begin{block}{Definition}
+\vspace{-15pt}
+\begin{align*}
+B(G)_{ij}&=1&&\Leftrightarrow&&\text{Kante $j$ endet in Knoten $i$}\\
+A(G)_{ij}&=1&&\Leftrightarrow&&\text{Kante zwischen Knoten $i$ und $j$}
+\end{align*}
+\end{block}
+\end{column}
+\begin{column}{0.48\textwidth}
+\begin{center}
+\begin{tikzpicture}[>=latex,thick]
+
+\def\dy{0.48}
+\def\dx{0.54}
+
+
+\begin{scope}
+\fill[color=red!20] (1.8,1.8) rectangle (4.75,2.15);
+\fill[color=darkgreen!40,opacity=0.5] (4.46,0.36) rectangle (4.79,2.65);
+\foreach \y in {1,...,5}{
+	\node[color=gray] at (5.3,{2.45-(\y-1)*\dy}) {\tiny $\y$};
+}
+\foreach \y in {1,...,6}{
+	\node[color=gray] at ({1.92+(\y-1)*\dx},2.90) {\tiny $\y$};
+}
+\draw[color=gray] (1.8,2.75) -- (4.7,2.75);
+\draw[color=gray] (5.2,2.55) -- (5.2,0.45);
+\node[color=gray] at ({1.92+2.5*\dx},3.1) {\tiny Kanten};
+\node[color=gray] at (5.3,{2.45-2*\dy}) [above,rotate=-90] {\tiny Knoten};
+\end{scope}
+
+\begin{scope}
+\fill[color=red!20] (1.8,-1.16) rectangle (4.25,-0.77);
+\fill[color=red!20] (2.3,-2.6) rectangle (2.63,-0.29);
+\foreach \y in {1,...,5}{
+	\node[color=gray] at (4.7,{-0.5-(\y-1)*\dy}) {\tiny $\y$};
+	\node[color=gray] at ({1.92+(\y-1)*\dx},-0.1) {\tiny $\y$};
+}
+\draw[color=gray] (1.8,-0.22) -- (4.2,-0.22);
+\draw[color=gray] (4.6,-0.4) -- (4.6,-2.55);
+\node[color=gray] at ({1.92+2*\dx},0.1) {\tiny Knoten};
+\node[color=gray] at (4.7,{-0.5-2*\dy}) [above,rotate=-90] {\tiny Knoten};
+\end{scope}
+
+\node (0,0) [right] {$\displaystyle
+\begin{aligned}
+B(G)
+&=
+\begin{pmatrix}
+1&0&0&1&1&0\\
+1&1&0&0&0&1\\
+0&1&1&0&1&0\\
+0&0&1&0&0&0\\
+0&0&0&1&0&1
+\end{pmatrix}
+\\[12pt]
+A(G)
+&=
+\begin{pmatrix}
+0&1&1&0&1\\
+1&0&1&1&0\\
+1&1&0&1&0\\
+0&1&1&0&0\\
+1&0&0&0&0
+\end{pmatrix}
+\end{aligned}$};
+
+\end{tikzpicture}
+\end{center}
+\end{column}
+\end{columns}
+\end{frame}
+\egroup