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Diffstat (limited to '')
| -rw-r--r-- | vorlesungen/slides/8/Makefile.inc | 1 | ||||
| -rw-r--r-- | vorlesungen/slides/8/chapter.tex | 1 | ||||
| -rw-r--r-- | vorlesungen/slides/8/inzidenzd.tex | 145 |
3 files changed, 147 insertions, 0 deletions
diff --git a/vorlesungen/slides/8/Makefile.inc b/vorlesungen/slides/8/Makefile.inc index 903fce2..68e2e27 100644 --- a/vorlesungen/slides/8/Makefile.inc +++ b/vorlesungen/slides/8/Makefile.inc @@ -9,5 +9,6 @@ chapter8 = \ ../slides/8/graph.tex \ ../slides/8/grad.tex \ ../slides/8/inzidenz.tex \ + ../slides/8/inzidenzd.tex \ ../slides/8/chapter.tex diff --git a/vorlesungen/slides/8/chapter.tex b/vorlesungen/slides/8/chapter.tex index 23d9aac..6ac3e98 100644 --- a/vorlesungen/slides/8/chapter.tex +++ b/vorlesungen/slides/8/chapter.tex @@ -7,3 +7,4 @@ \folie{8/dgraph.tex} \folie{8/grad.tex} \folie{8/inzidenz.tex} +\folie{8/inzidenzd.tex} diff --git a/vorlesungen/slides/8/inzidenzd.tex b/vorlesungen/slides/8/inzidenzd.tex new file mode 100644 index 0000000..19da619 --- /dev/null +++ b/vorlesungen/slides/8/inzidenzd.tex @@ -0,0 +1,145 @@ +% +% inzidenzd.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\definecolor{darkgreen}{rgb}{0,0.6,0} +\begin{frame}[t] +\frametitle{Inzidenz- und Adjazenz-Matrix} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.40\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,color=darkgreen] (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]; +\fill[color=red!20] (B) 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[color=darkgreen] at ($0.4*(B)+0.6*(D)$) [left] {$\scriptstyle 7$}; + +\end{tikzpicture} +\end{center} +\vspace{-15pt} +\begin{block}{Definition} +\vspace{-20pt} +\begin{align*} +B(G)_{ij}&=-1&&\Leftrightarrow&&\text{Kante $j$ von $i$}\\ +B(G)_{kj}&=+1&&\Leftrightarrow&&\text{Kante $j$ nach $k$}\\ +A(G)_{ij}&=\pm 1&&\Leftrightarrow&&\text{Kante von $i$ nach $j$} +\end{align*} +\end{block} +\end{column} +\begin{column}{0.58\textwidth} +\begin{center} +\begin{tikzpicture}[>=latex,thick] + +\def\dx{0.84} +\def\dy{0.48} + +\begin{scope}[xshift=4cm,yshift=3cm] +\fill[color=red!20] +({-0.67-(7-1)*\dx-0.4},{-0.38-(2-1)*\dy-0.2}) +rectangle +({-0.67-(7-7)*\dx+0.2},{-0.38-(2-1)*\dy+0.16}); +\fill[color=darkgreen!40,opacity=0.5] +({-0.67-(7-7)*\dx-0.4},{-0.38-(5-1)*\dy-0.2}) +rectangle +({-0.67-(7-7)*\dx+0.2},{-0.38-(1-1)*\dy+0.16}); +%\draw (0,0) circle[radius=0.05]; +\foreach \x in {1,...,7}{ + \node[color=gray] at ({-0.67-(7-\x)*\dx},0.0) {\tiny $\x$}; +} +\draw[color=gray] ({-0.72-6*\dx},-0.1) -- (-0.6,-0.1); +\foreach \y in {1,...,5}{ + \node[color=gray] at ({0},{-0.38-(\y-1)*\dy}) {\tiny $\y$}; +} +\draw[color=gray] (-0.1,-0.28) -- (-0.1,-2.4); +\node[color=gray] at ({-0.67-(7-4)*\dx},0.04) [above] {\tiny Kanten}; +\end{scope} + +\begin{scope}[xshift=2.32cm,yshift=-0.24cm] +%\draw (0,0) circle[radius=0.05]; +\fill[color=red!20] +({-0.67-(5-1)*\dx-0.4},{-0.38-(2-1)*\dy-0.2}) +rectangle +({-0.67-(5-5)*\dx+0.2},{-0.38-(2-1)*\dy+0.16}); +\fill[color=red!20] +({-0.67-(5-2)*\dx-0.4},{-0.38-(5-1)*\dy-0.2}) +rectangle +({-0.67-(5-2)*\dx+0.2},{-0.38-(1-1)*\dy+0.16}); +\foreach \x in {1,...,5}{ + \node[color=gray] at ({-0.67-(5-\x)*\dx},0.0) {\tiny $\x$}; +} +\draw[color=gray] ({-0.72-4*\dx},-0.1) -- (-0.6,-0.1); +\foreach \y in {1,...,5}{ + \node[color=gray] at ({0},{-0.38-(\y-1)*\dy}) {\tiny $\y$}; +} +\draw[color=gray] (-0.1,-0.28) -- (-0.1,-2.4); +\node[color=gray] at ({-0.67-(5-3)*\dx},0.04) [above] {\tiny Knoten}; +\node[color=gray] at ({0.00},{-0.38-(3-1)*\dy}) + [above,rotate=-90] {\tiny Knoten}; +\end{scope} + +\node at (0,0) {$\displaystyle +\begin{aligned} +B(G) +&= +\begin{pmatrix*}[r] +-1& 0& 0& 0& 1&-1& 0\\ + 1&-1& 0& 0& 0& 0&-1\\ + 0& 1&-1& 0& 0& 1& 0\\ + 0& 0& 1&-1& 0& 0& 1\\ + 0& 0& 0& 1&-1& 0& 0 +\end{pmatrix*} +\\[20pt] +A(G) +&= +\begin{pmatrix*}[r] + 0&-1&-1& 0&-1\\ +-1& 0&-1&-1& 0\\ +-1&-1& 0&-1& 0\\ + 0&-1&-1& 0&-1\\ +-1& 0& 0&-1& 0 +\end{pmatrix*} +\end{aligned}$}; +\end{tikzpicture} +\end{center} +\end{column} +\end{columns} +\end{frame} |